REPORT UDC 551.464:551.465 CM-71 August 1987
ON INVERSE METHODS FOR COMBINING CHEMICAL AND PHYSICAL OCEANOGRAPHIC DATA: A STEADY-STATE ANALYSIS OF THE ATLANTIC OCEAN
Bert Bolin, Anders Björkström, Kim Holmen and Berrien Moore
Institute for the Study of Earth, Oceans and Space, University of New Hampshire, Durham, New Hampshire 03824 USA, and Laboratoire de Physique et Chimie Marines, Université Pierre et Marie Curie, Paris VI.
DEPARTMENT OF METEOROLOGY <5 Ä'J> UNIVERSITY OF STOCKHOLM g ^f$f £ INTERNATIONAL METEOROLOGICAL -7, *f S> INSTITUTE IN STOCKHOLM
ISSN 0280-445X DEPARTMENT OF METEOROLOGY REPORT CM-71 1 (220) UNIVERSITY OF STOCKHOLM (MISU) 1987-08-01 UDC 551.464:551.465 INTERNATIONAL METEOROLOGICAL INSTITUTE IN STOCKHOLM (IMI)
Arrhenius Laboratory S-106 91 STOCKHOLM, Sweden
Tel 08/16 2406
ON INVERSE METHODS FOR COMBINING CHEMICAL AND PHYSICAL OCEANOGRAPHIC DATA: A STEADY-STATE ANALYSIS OF THE ATLANTIC OCEAN by Bert Bolin, Anders Björksträn, Kim Holmen and Berrien Moore *)
Abstract An attempt has been made to increase the spatial resolution in the vise of inverse methods to deduce rates of water circulation and detritus formation by the simultaneous use of tracer data and the condition of quasi-geostrophic flow. It is shown that an cverdetermined system of equations is desirable to permit analysis of the sensitivity of a solution to errors in the data fields.
The method has been tested for the Atlantic Ocean, in which case we employ an 84-box model (eight layers in the vertical and 12 regions in the horizontal define the box configuration). As quasi-steady tracers we consider dissolved inorganic carbon (DIC), radiocarbon, alkalinity, phosphorus, oxygen, salinity and enthalpy. The condition of quasi-geostrophic flow is employed for flow across all vertical surfaces between regions except close to the equator. Water continuity is required to be exactly satisfied by the use of a set of closed loops to describe the advective flow. The method of least squares is used to derive a solution and in . . so doing we also require, that turbulence only transfers matter down tracer gradients (i.e. eddy diffusivity is nonnegative) and that detritus is formed only in surface boxes and is destroyed in the water column below.
It is shown how appropriate weighting of the equations in the set is decisive for the solutions that we derive and that great care must be taken to ascertain that the interior tracer distributions and the boundary conditions in terms of exchange of tracer material with the exterior are compatible. The enthalpy equations turn out not to fulfil such a demand and have not been used in deriving the solutions presented in the paper.
A basic solution with an ageostrophic flow component of about 15% is derived and compared with current knowledge about the general circulation of the Atlantic Ocean and the rates of detritus formation
Institute for the Study of Earth, Oceans and Space, University of New Hampshire, Durham, New Hampshire 03824, USA, and Laboratoire de Physique et Chimie Marines, Université Pierre et Marie Curie, Paris VI. and destruction. ïhe sensitivity of the solution to uncertainties in the data field is presented. It is shown that the solution is markedly dependant on whether turbulent transfer is included or not. On the other hand, we are not able to détermine how important boundary conditions on 14 exchange of 002 and C between the atmosphere and the sea are for our solution and accordingly for the uptake of present excess in the atmosphere. Nor is it possible to determine with any degree of confidence which of different sets of Fedfield ratios would best ss.tisfy the present models for detritus formation and destruction. Also the most likely rate of deep water overturning cannot be determined very accurately. Ihe uncertainty of current estimates found in the literature is also considerable.
Some of the difficulties arise from a mismatch between the questions asked and the resolution of the data; an example is the turnover time for individual deep ocean boxes versus the small gradients in the corresponding 14C data. In other cases, the structure of the model with its large boxes does not take full advantage of the information content in the data. We conclude with some observations about future research efforts that might successfully address these difficulties. CONTENTS 3
page
List of symbols 6
1. Introduction 12
Part I THE THEORETICAL BASIS 18
2. Overview 18
3. Diagnostic equations 23
3.1. The general setting 23
3.2. The condition of geostrophy 26
3.3. Water continuity definition of "loops" as
advective variables 28
3.4. Continuity equations for tracers 33
3.5. The set of diagnostic equations and the
inequality constraints 36
4. Analysis of matrix characteristics and principles
for solution 39
4.1. Basic characteristics of the system 39
4.2. Equality constraints 44
4.3. Weighting and scaling 45
4.4. Inequality constraints and noise in the data 47
5. Derivation of a solution 48
Bart II THE DATA BASIS 57
6. Overview 57
7. Box topology 58
8. Data 65
8.1. Geostrophy 65
8.2. Tracer distributions 68
8.3. Boundary conditions 72 4
8.3.1. Water 72
8.3.2. Exchange of tracer material and enthalpy
with adjacent seas 76
8.3.3. Air-sea exchange of heat, carbon dioxide,
radiocarbon and oxygen 77
Part III REALIZATION OF THE MODEL FOR THE ATLANTIC OCEAN 81
9. Overview 81
10. The loops: Construction of the matrix L and the
vector IDQ 81
11. Construction of the matrix A and the vector b 83
12. The unweighting: Construction of matrices Dz and Dg 88 13. The inequality constraints: Construction of a matrix G
and vector h 92
Part TV RESULTS 95
14. Overview 95
15. A set of basic experiments 98
15.1. Some introductory remarks 98
15.2. The quasi-geostrophic condition 102
15.3. A reference solution 110
15.3.1. The field of advective flow 110
15.3.2. The field of turbulent exchange 118
15.3.3. New primary production and detritus flux 120
15.4. The question of indeterminancy of the
referance solution 123
16. Sensitivity studies 125
16.1. Sensitivity to external forcing 127 5
16.1.1. Water flux boundary conditions 127
16.1.2. Radiocarbon decay 129
16.1.3. Carbon dioxide exchange 131
16.2 Sensitivity to tracer data uncertainties 134
16.3. Sensitivity to model assumptions 139
16.4. Error analysis of the base case 141
16.5. The steady-state assumption and representativity
of data 151
16.5.1. Scale considerations 151
16.5.2. Do available data resolve the processes
considered? 158
16.5.3. The problem of data on dissolved inorganic
carbon 162
16.6 Conclusions of the sensitivity studies 163
Issues in Methodology 165
17.1. Introduction 165
17.2. Incompatibilities due to inaccurate data:
Total Least Squares 166
17.3. Incompatibilities due to approximate equations 169
17.4. The combined effect of inaccurate data and
approximate equations 170
17.4.1. Separable Least Squares:
An experimental approach 170
17.4.2. Iterative minimization 171
17.4.3. Singular value decomposition:
A classical approach 173
Conclusions 176
Acknowledgements 183
References 184 List of symbols
Physical chemical and biological variables in physical space x, v. z. a (unit.) : years D decay of C in a reservoir E net flux of tracer to adjacent reservoir f Coriolis parameter. f j .i flux of tracer n from reservoir (box) i to j. F withdrawal of carbon due to new primary production or release from net decay.
Fc withdrawal of carbon due to new carbonate formation or release due to dissolution.
F_.jai', F.jxa_ flux across the airsea interface to or from reservoir i. F^ flux from the exterior across an external boundary. G accumulation of tracer in reservoir. K eddy diffusivity. m mass flux of water. m mass flux of water at reference level z Mg horizontal flux of water across vertical surface in layer s. superscript n indicates tracer, 1,2,...N. n = C carbon = * radiocarbon ( C) = A carbonate alkalinity = O oxygen = P phosphorus = S salinity = T temperature 7 potential temperature partial pressure of carbon dioxide in sea water. water pressure. amount of tracer matrial (tracer n) per unit mass of sea water. margin of uncertainty around q11
(unit) : metric tons velocity in the horizontal direction x. reference level. surface between layers s and s+l.
Redfield ratio for compound n in the process of primary production.
Redf ield ratio for compound n in the process of carbonate formation. distance between center points of adjacent reservoirs
(boxes). norm of residual vector. radioactive decay constant for tracer n water density. direction perpendicular to surface (Sh) between adjacent boxes. surface between adjacent boxes.
number of reservoirs, "boxes".
number of closed loops that define water continuity.
number of horizontal regions, into which the ocean
domain is divided.
number of layers into which the ocean domain is 8 divided.
t = 1, 2 ... T: number of interfacial surfaces between boxes.
Tn: number of vertical interfacial surfaces
Ty: number of horizontal interfacial surfaces.
w = 1, 2 ... W: number of interfacial surfaces between R regions.
Vectors, matrices and variable spaces.
A General symbol for coefficient matrix in a
minimization problem.
A+ Pseudo-inverse of A.
AQ Submatrix of A that only includes water continuity
equations.
A^ Matrix A after truncation of small singular values.
A' Matrix A constructed from perturbed data.
A , A , A , or A*1 where n is a tracer index. See text at equation
(11.1).
£(A, b)J symbol used as abbreviation for '•minimize li Ax-bl|"
$ (A, b), (6, h)J abbreviation for "minimize |j Ax-b| subject to Gx. > h"
B space of vectors b
1^ space of vectors b^
b vector composed of sources and sinks of tracers at
the external boundary of the domain and sink due to
radioactive decay,
b^ vector defined as Dg (b - AxQ) (see these symbols)
bn sub-vector of b, including only the equations for
tracer n.
bF1 vector composed of water exchanges at external
boundaries.
E^ space of vectors tP.
D general symbol for a diagonal matrix. 9
"equilibration" or "urweighting" matrix of size
Jb x Jb' used *° eliminate unintentional weighting in {(A, b)] unweighting matrix for Z.
Modification matrix to ascertain detritus continuity
(Part III).
Perturbation matrix for A (Part IV).
general symbol for matrix of inequality constraints.
where n is a tracer index: See Section 17.4.
space of vectors h
vector composed of right hand sides in inequality
constraints,
identity matrix.
number of rows in A, also equal to number of
components in b.
number of columns in A, also equal to number of
components in x.
Kernel matrix (Part I).
Square matrix (Part IV) with I rows and I columns,
constructed so that Kn« q + k" is an algebraic
reformulation of A^K - bn, where if1 and k?1 are
independent of q.
vector with I components (see K0).
subspace of M, including all m such that &ja = 0 matrix for transformation from Z to X.
matrix with T rows and T-(I-l) columns constructed
so that each of its T-dimensional column vectors
describes a closed circulation, a "loop", when
interpreted as an m-vector.
Pseudo-inverse of LQ. space of vectors m. 10 m T-diniensional vector composed of mass fluxes (water) through each of the T interfacial surfaces, m Smallest-norm solution to A^ = h N Non-linear operator from Z to X. q Vector composed of all tracer concentrations r Residual vector in B.
S Subset of M containing all m such that m = m0 + m^ where m^ is in K. S Rectangular matrix containing the singular values of A on the main diagonal, fl^j. Matrix S with some singular values replaced by zeroes (truncation). T diagonal transfer matrix that describes the amount of C dissolved in a reservoir when one unit of C is supplied by decomposition or dissolution.
U (Jb x Jb)-matrix consisting of singular vectors in B. UQ Jjj-dimensional singular vector in B.
V (Jx x Jx)-matrix consisting of singular vectors in X. v Jx-dimensional singular vector in X.
W Diagonal (Jx x Jx) weighting matrix for Total least Squares solution. x General symbol for solution vector. xQ Vector m0 with Tf2*I zeros attached. Xj^ Solution vector when tracer n is upweighted. x^ Solution vector in unweighted space. X Space of vector x.
X^ Space of vectors Xu. X Representation of an m-vector as a linear combination of loops, i.e., m = L^. Y Space of vectors y. z Same vector as x, but the advective components are 11
replaced by a vector y. z Modification of z to ascertain detritus continuity. zu Vector Z in an unweighted space. Z Space of vectors z.
Zu Space of vectors z^. 12
1 Introduction
Observations of oceanic currents are not yet adequate for producing a reliable quantitative description of the gross features of the circulation of the ocean, particularly not at depth. The fact that direct current observations are not representative enough to be particularly useful is especially troublesome. Constant density floating buoys have been designed to overcame this difficulty, but such data are sparse. Still we gradually have obtained a reasonably good picture of the major ocean currents in the surface layers of the sea and it has also been possible to establish the existence of currents in the deep sea by direct measurements, e.g. the coastal current underneath the Gulf
Stream flowing in the opposite direction. Our present knowledge of the general circulation of the ocean is, however, to a very considerable degree based on hydrographie measurements, i.e. temperature and salinity and the systematic use of the assumption of quasi-geostrophic flow. As is well-known it then is necessary to specify a level of no motion or to prescribe the flow at some level in order to derive the field of horizontal motions. For this reason considerable uncertainty has prevailed with regard to the precise interpretation of such analyses.
The method also fails close to the equator because the assumption of quasi-geostrophic flow is not valid. Further, the vertical motions remain undetermined except insofar as constrained by water continuity.
Still the approach has become the classical one to derive the large-scale flow of the oceans since the pioneering work by Wust (1935) and others in the 1930's.
The analysis of hydrographie data has recently been advanced by imposing additional dynamical constraints. Stammel and Schott (1977) have 13 developed a theory, termed the Beta Spiral, which is based on the assumption that the divergence of the horizontal component of the velocity can be estimated with the aid of the geostrophic relation, and that thereby a reference velocity can be determined. The method has been vised for extensive studies of the North Atlantic circulation pattern by
Olbers et al. (1985). Wunsch (1978) and Fiaderio and Veronis (1982) impose the Jcinematical constraint of water continuity within isopycnal layers. In this case, however, the problem usually remains underdetermined, and some minimization principle has to be employed.
Further, the assumptions of approximating the horizontal divergence with that of the geostrophic velocity (Beta Spiral) and of isopycnic flow does not permit us to deduce the three-dimensional flow pattern adequately.
Wunsch and Minster (1982) have developed a diagnostic method by using the quasi-steady distribution of temperature and salinity assuming conservation of heat and salt in the interior of the ocean and employing the technique of matrix inversion under inequality constraints.
In addition to the transfer by advective flux it becomes necessary to consider transfer by turbulent processes, which can be done most simply in terms of the concept of eddy diffusivity. Using only two tracers and the requirement of water continuity the problem is urKierdetermined.
Wunsch and Minster solved the problem by a classical application of the pseudo-inverse i.e. by minimizing the norm of the solution vector. In a later paper but still in an underdetermined setting, Wunsch (1985) has deduced the range of possible solutions which are consistent with a set of kinematic and sometimes qualitative constraints.
A fundamentally different approach to resolve this problem is sought in the development of general ocean circulation models, with the aid of which the steady-state circulation of the oceans may be deduced in 14 response to a given external forcing i.e., specification of sources and sinks at the ocean surface of momentum, energy and water. In a long-term perspective this undoubtedly is the most rational approach to improving our understanding of the general circulation of the oceans. Tracer distributions, steady-state and transient, then also can be used for independent validation of such dynamical computations. This approach will, however, not be considered further in the present context.
Still another approach to studying the general circulation of the ocean has been taken by the chemical oceanographers. They ask the question:
What are the necessary transfer processes due to water motions in order to explain the quasi-steady distribution of chemical compounds in the ocean? What can we learn about the biochemistry of the sea by the
simultaneous use of several tracers? How do we make use of the transient tracers, particularly tritium, 14C and chlorofluorocarbons that now are
invading the world oceans due to man-made emissions into the atmosphere?
The basic idea is to make use of the fact that the transfer of matter by water motions (advective and turbulent) affects the various elements
similarly and that accordingly the different observed distributions are
due to different distributions of sources and sinks.
The recognition that cosmic rays produce the radioactive isotope ^4c and
that the distribution of C in the sea could be used to determine the
rate of renewal of the deep waters of the oceans led to the development
of simple ocean models to interpret available 4C data (Craig, 1957).
The simultaneous use of C and more traditional marine tracers was
first attempted by Bolin and Stommel (1961) and further developed by
Keeling and Bolin (1967). These early developments were based on very
simple (usually 2-box) models. Oeschger et al. (1975) recognized the
need to resolve the vertical structure of the oceans in more detail to
depict reality better by formulating the box-diffusion model and 15
Siegenthaler (1983) further elaborated on this same idea by also
considering the isopycnal ventilation of the deep sea through the cold
polar regions (the out-crop model). Peng and Broecker (1982) have also
made extensive use of tracers in the study of the biogeochemistry of the
oceans. Their concept of large-scale mixing of major water masses ("end
members") may also be used to interpret some aspects of the observed
distributions. Some of these analyses may, however, be criticized
because in formulating the simple models implicit assumptions are made
which can significantly influence the results. These are, therefore,
often not conclusive.
Much more care must be exercised in using such methods for analyses of
the interplay between ocean circulation and biochemical processes in the
sea. A more rigorous method of analysis has been presented by Bolin et
al. (1983). The simple structure of the model (a set of reservoirs into which the ocean is divided) and physical, chemical and biological
processes to be considered were carefully defined. An initial steady
state was assumed and a sufficient number of compounds (tracers) were
considered to overdetermine the problem of how water motions and
biochemical processes interact in establishing the observed
distributions.
Bolin et al. (1983) employed five tracers in their study of the global
ocean circulation, namely total dissolved inorganic carbon (Die),
alkalinity (A), radioactive carbon ( C), oxygen (0), and dissolved
inorganic phosphorus (P) and imposed the condition of water continuity.
A very coarse resolution model was used, i.e. merely 12 reservoirs for
the global ocean. The fluxes of organic and inorganic detrital matter
from the surface layers into the thermocline region and the deep sea was
considered, and these fluxes became additional unknowns. The source and
sink terms for the biological important tracers were interrelated by 16 assuming constant values for the Redfield ratios. No deposition of detrital matter on the sea bottom was permitted. Since altogether five tracers, continuity of water and of detrital matter were used, the inverse problem was formally overdetermined, and a condition of idnimizing the errors of the conservation equations was employed as well as a set of inequality constraints requiring that turbulent transfer be along the concentration gradient of the tracers and that detrital matter forms in surface reservoirs and is dissolved or decomposed in deeper layers of the ocean. The deduced circulation pattern generally agreed with the current view of the gross features of the global ocean circulation.
It is appropriate here to refer also to the elaborate study of a very similar kind by Riley (1951). A revaluation of this work is warranted since extensive experimentation can be carried out using available powerful computors and much better data are now available.
The question naturally arises if the different inverse approaches can be combined. How can we best utilize the information contained in the distribution of tracers in the sea to improve on the physical relationships presently used to deduce the fields of water motion?
Conversely and of equal fundamental importance: How can we better use present knowledge about the general circulation of the oceans to learn about the biochemistry of the sea? In other words, can we unify the inverse methodologies for ocean circulation and biochemical diagnosis referred to above? An analysis of this question is one of our prime objectives.
As will be seen from the following presentation, considerable difficulties will be encountered in attempting to solve this problem with better spatial resolution than has earlier been employed by 17 chemical oceanographers and for example used by Bolin et al. (1983).
Although we shall in this way be able to deduce the water circulation in a major ocean (the Atlantic Ocean) and the spatial patterns of detrital matter formation, decomposition and dissolution, it will also became very clear that our prime results are of a methodological nature. We shall therefore adopt the more specific objective of this paper to be:
How well can we determine the interplaY of physical, chemical and biological processes in the sea from the simultaneous observations of the quasi-steady distributions of oceanic tracers? It will be of particular interest to analyze the importance of:
- the spatial structure of the model being employed
- the spatial scale adopted in the model in relation to that of the
processes being described
the adequacy of the processes being considered in the model.
The following presentation has been organized in such a manner that it should be possible to appreciate the method of approach (Part I; sections 2 and 3) and its application to a study of the Atlantic Ocean
(Parts II and IV) without necessarily considering in detail the techniques of matrix inversion under inequality constraints (Part I sections 3 and 4) and the technical realization of our method of solution (Part III). 18
PART I THE THEORETICAL BASIS
2 Overview
As a starting point we adopt the classical assumption that outside a rather narrow zone around the equator the horizontal motions in the ocean are quasi-geostrophic. We also formulate the problem in such a way that we are able to analyze how our final diagnosis of the flow patterns and turbulent mixing is dependent on the degree to which the condition of geostrophy is being satisfied.
Water continuity within the oceanic domain under consideration remains a fundamental requirement. As was the case for our 12 box model and as will be discussed in detail later, our current model is formulated as an overdetermined (or rather incompatible) problem, i.e. our basic diagnostic equations are not satisfied exactly. In an werdetermined framework one, in principle, may accept solutions that do not satisfy the requirement of water continuity exactly. This possibility was observed by Wunsch and Minster (1982) and is currently being explored by one of us (BM) in the context of direct-indirect problem as formulated by Fiadeiro and Veronis (1984). However, no observations are required in the way we formulate our condition of water continuity. The use of tracer equations, on the other hand, is dependent on knowledge of the tracer concentration distributions, and these are only approximately determined. Secondly, after having derived an optimum transfer pattern we may wish to use our solution to compute the transient behaviour of certain tracers that have not been used in the derivation of the flow pattern. The recent observations of tritium and chloroflurocarbons as well as bomb carbon-14 may be particularly useful in this regard. It is necessary, however, for such computations to be of interest that the 19 water continuity be exactly satisfied, since otherwise, artificial sources and sinks of tracers are implicitly introduced. We accordingly develop a method for satisfying the water continuity requirement exactly and, as we shall see, this method presents other advantages as well.
Since the conditions of geostrophy and water continuity alone leave the resolution of the ocean circulation underdetermined, we employ a set of tracer equations to add further information to the determination. We pay the price, however, of also having to consider the importance of subgrid scale turbulent processes and the role of detrital flux. In either case but particularly in being able to treat new primary production and decomposition this price buys important information about the global carbon cycle. (New primary production is the net primary production that leaves the photic zone; hence, it is a measure of the rate at which carbon is removed from the photic zone and falls to deeper layers in the form of detrital material. See Eppley and Peterson, 1979). As a matter of fact, and as referred to before, our initial attempt to analyze by matrix inversion techniques the global ocean circulation, including turbulent processes, was due to this concern with regard to the role of the oceans in the carbon cycle.
Any theoretical treatment of a problem concerning our environment, such as the present one of deducing the oceanic transfer by inverse methods, must be based on some kind of model. In reality, of course, the natural phenomena are so complex and our data so few that all such problems are underdetermined. However, only by adopting models, and thereby, by definition, imposing a reduction on the complexity of reality do overdetermined systems arise, and also only by adoption of models can we hope to make efficient use of the available data and information. It is obvious that whether or not the results will be of interest depends on how well our model captures the essence of the phenomena in nature which 20 it seeks to describe. Much too little attention has been devoted to this irrportant aspect of the analysis, and considerable uncertainty may stem primarily from an inadequate realization of reality in adopting a particular structure for the model.
Assuming for the moment that we are able to design a physically- chemically "reasonable" model, we face the question of how detailed we may assume it to be. Basically, the number of degrees of freedom of a model should not exceed the degrees of freedom that the data contain, which is used to determine the set of unknowns that describe the circulation, the transfer processes and the biochemical processes.
Wunsch (1985) has argued, however, that the problem could well be underdetermined and rather than search for a single solution, one should explore the range of values for solutions and their implication for other geophysical phenomena. It should be recognized, however, that in such an analysis one does not consider explicitly inperfections or inconsistencies in the model. Rather, one assumes implicitly that all available data fit the model exactly and that the range of uncertainty is dependent exclusively on insufficient data. Even if direct measurements, from which model data are derived, are accurate, model variables often represent mean values in space and/or time and cannot necessarily be obtained accurately from available measurements.
It, therefore, seems appropriate to design our diagnostic model with sufficient simplicity that it formally becomes overdetermined. Since, in fact, any number of data is insufficient to describe reality in all its complexity and since the problem accordingly in reality is not overdetermined, it seems preferable to call such a model incompatible, i.e. the model and the data are incompatible because of an inadequate model or inaccurate data or both. Having obtained an optimum solution (a best fit solution) it is, however, important to analyze its sensitivity 21
to errors in the data field on which it is based. It is well-known that results from the inversion of large matrices may be quite sensitive in this regard (Fiadeiro and Veronis, 1984), particularly if the system is not properly unweighted and scaled. In addition to the problem of unweighting and scaling the equations, we need to explore the sensitivity of the solution to noise in the data. As part of this exploration, we describe an iterative procedure; whereby, it may be possible to determine how the system might be satisfied better by also systematically altering the basic equations as permitted within the noise of the data.
It should be mentioned that a system can be incompatible even in the case when the number of unknowns exceeds the number of defining equations. This may not occur often in practice with small systems, when it is simple to grasp the overall characteristics of the problem and thereby avoid such configurations. It is, however, not uncommon to encounter large complex systems that are incompatible although formally underdetermined. For this and other reasons it may be best to call urderdetermined systems simply indeterminate•
Thus there are four possible formulations: a) Compatible and determinate (one and only one exact solution); b) Compatible and indeterminate (an infinite number of exact solutions); c) Incompatible and determinate (one solution to the nonzero least squares approximation); and d) Incompatible and indeterminate (an infinite number of solutions that solve the (nonzero) least squares approximation). 22
In summary, we address the problem of deducing for an ocean basin the advective circulation, rates of turbulent transfer (in terms of eddy diffusivities), new primary production of organic tissue and its decay as well as new carbonate formation and its dissolution by assuming simultaneously that:
1) the advective flow is quasi-geostrophic,
2) the water continuity is exactly satisfied, and
3) a set of tracer equations are approximately satisfied.
The problem is formulated in terms of a formally incompatible
(cverdetermined) system subject to a set of inequality constraints which arise, in part, due to requirements that eddy diffusivities are nonnegative and that new primary production and carbonate formation only take place in surface layers. It is solved by inverse methods.
The method is used for a preliminary analysis of the Atlantic Ocean.
Since it is well-known that the solutions to large systems of linear equations may be sensitive to small errors, careful verification of our results using independent data is essential. Principally all hydrographie data and all data for steady state distribution of tracer
fields are used in formulating our basic set of equations and cannot be used for verification. To the extent that direct observations of large
scale ocean water transfer are available, these will be of great
importance. Few such data from the open sea exist, however. Direct measurements of new primary production of organic and carbonate matter, as well as detrital flux into the deep sea are still quite crude and not
always representative, but may of course be useful.
Data are also available on transient tracers in the sea.
The production of tritium and 14C in nuclear tests and chlorf lurocarbons by the chemical industry have provided three tracers that are invading 23 the sea. The steady-state solution for transfer processes that we derive can be tested by determining the way it would describe such an invasion and quite extensive measurements are available against which comparisons can be made. Our analysis, however, aims at reproducing the large-scale flow in the Atlantic Ocean, as averaged over at least several hundreds of years. Whether or not a model is succesful in this regard can hardly be evaluated on the basis of how well it reproduces the uptake and spread of a transient tracer, the emissions of which have taken place during a short period of time, say the last thirty years. The successive invasion of a transient tracer may proceed by yearly pulses, or by small and meso-scale eddies that are not being resolved individually. Failure to reproduce those processes does not exclude the possibility that the steady-state model could be a reasonable first approximation as such.
To use transient data appropriately in this way requires careful consideration of the continuity equation without neglect of the time derivate. The volume integrations that are undertaken in order to form box averages may need to be reanalyzed. Therefore, in the present study, we only use data on tracers where a steady-state assumption can be made, but intend to return to transient integrations in the near future.
3 Diagnostic equations
3.1 The general setting.
We consider an ocean basin and seek to determine its pattern of circulation, the rates of turbulent transfer and biochemical processes.
Our approach is to apply a set of diagnostic equations and employ data on the bulk properties of ocean water, specifically the spatial, quasi-steady state distributions of temperature, salinity and a set of biologically active tracers. We also shall need to prescribe boundary 24 conditions for the ocean domain that is being considered related to the exchange of heat, water, and chemical properties with the atmosphere and adjacent aquatic bodies.
The analysis will be formulated through a set of finite-element equations or, in other words, in terms of a set of equations for specific subregions of the domain, i.e., "boxes". The diagnostic equations will be obtained by assuming quasi-geostrophic flow (section
3.2), water continuity (section 3.3) and tracer continuity (section
3.4).
In order to illustrate the handling of a large number of boxes and for a discussion of the concepts of indeterminacy and incompatibility, it is useful to be able to refer to a structurally simple model that is similar to the one which we later use in our analysis of the Atlantic
Ocean.
Consider an ocean domain which consists of R (=12) horizontal regions and S (=8) layers in the vertical, i.e., the total number of boxes is
I=R*S (=96). (Figure 1) (for an océanographie application, see Figure
8). It is readily seen that this configuration has Th=W*S (=136) vertical interfacial surfaces through which there is horizontal transfer and TV=R*(S-1) (=84) horizontal interfacial surfaces through which there is vertical transfer; W (=17) is the number of surfaces between the R regions. There is accordingly T==Tn+T!v.(=220) interfacial surfaces. In addition, some of the boxes represent boundaries between the ocean domain and the atmosphere, adjacent oceanic basins and other aquatic bodies. Layers 1 2 3 ai 4 5 6 7 8
Fiqure i. Geometrical arrangement of boxes in the domain under consideration. ïhe bold-face numerals indicate the regions (R = 12) and the thin numerals in the uppermost layer (1 to 17) refer to the înferfaces between regions. (W = 17). ihere are s ( = 8) layers in the vertical. 26
3.2 The condition of geostrophv.
We consider first the condition imposed by geostrophy in the abstract,
ignoring for the moment that such conditions do not hold near the equator. In Figure 2, the rectangle ABCD denotes one of the W
interfacial surfaces in the model. The lines zs-1 and z define one of the S model layers. The velocity in the x-direction, u, and the density, / vary from place to place in ABCD. The product of these two, ro(Y/ z) = u(y, z) • ^ (y, z) is a mass flux. We seek an expression for the integral of m taken over the shaded area v&s in layer s between the
separating levels zs-1 and z&. For any two horizontal directions x and y, where y is 90 degrees to the left of x, the balance between Coriolis
acceleration and pressure gradient is
- f U - S 1^ -~0 (3-D where f is the Cöriolis parameter. Since hydrostatic conditions prevail, we have <^/ô2 - - <\ ] so that differentiation of (3.1) with respect to z gives
^tM-~^ (3.2)
We assume a reference level zQ (which can be a function of y) where the
mass flux is m0 (y) = m (y, zQ (y) ). Since f is independent of z we can integrate (3.2) in the vertical and get
2
The mass flux through the area -ils will be «» * )) *-y*-I)vM (\(H) «v * (3'4) 11
Figure 2. Vertical section along a wall between regions. For notations see text. 28
The last term on the right side can be evaluated numerically, given data on ) in the wall and having made a choice of reference level (in our case, the level z = 0 has been used). The first term on the right side can be integrated, using the fact that mQ is independent of z:
j) '^ocMî -- foo • J\s (3.5) A.s where Tttc is the average value of the water flux ( u~ )' ) along the reference level 20 . The S mass fluxes itig } s - \j, .^$ are thus tied together by S equations of the form
rv)s Ï"ST0 -_as *• Ms t s--\J..-/^S (3.6)
where -0.4 and Mj can be computed from hydrographie data. The average value wo becomes an additional unknown. Each of the W walls thus supplies S equations and one unknown.
Alternatively, the mass flux m0 can be eliminated between successive equations (3.6), in which case we obtain S-l equations at each wall for
the flux differences between successive layers:
m s-n - ),.--, AS'-I (3.7) n,s " -asti -°-s. AW, ' We shall use the latter formulation.
3.3 Water continuity; definition of "loops" as advective variables.
The continuity equation for water, i.e., the equation
(3.8)
where Fb denotes the local supply of water from external sources can be integrated over an arbitrary box; and gives
/,, sign (i,t) * mj. = b^ t - \}... ; i . (3.9) t* I 29
The technical function sign (i, t) is either plus one or minus one, depending on sign conventions that can be adopted arbitrarily. (If we further define sign (i, t) =0 when the flux m^. does not involve box i, the summation over t is formally correct).
The term b^ stands for the net source of water to box i from external regions. It may consist of exchange with adjacent ocean basins, or net precipitation, or both.
We combine the T mass fluxes m^. to a T-dimensional vector m, and also combine the I net sources L to an I-dimensional vector b111 . We then summarize the I continuity equations (3.9) in matrix form:
^•111 = b1" (3.10)
where the (I*T) -dimensional matrix AQ will be defined by
(Vit = sl9n (*•* t)
Since one of the water continuity equations is redundant, the rank of AQ is 1-1. With the aid of (3.10) we may derive by using the pseudo-inverse, AQ, of AQ (also called the Moore-Penrose inverse or the generalized inverse; see Wunsch, 1978; Fiadeiro and Veronis, 1982, or Bolin et al., 1983 (the Appendix)), a particular solution m with minimum norm. This solution, mQ, describes an advective field that satisfies exactly the boundary conditions b™ for the water exchange with the exterior.
Equation (3.10) constitutes a set of I equations that are to be part of the total set that defines our system. Treating these equations as simply part of the system would imply, however, that the water requirement would be only approximately satisfied since the overall system is incompatible. It would, of course, be possible to weight these 30
equations heavily in the search for a best fit solution and thereby
reduce the errors in the water continuity equations. As is pointed out
in section 4, it is desirable, however, to ascertain perfect water
continuity but also to avoid introducing weights.
Our approach to this problem is as follows. We first observe that any
solution m to equation (3.10) can be written uniquely as
m = mh + m0 (3.11) where SL is a solution to the homogeneous water continuity equations,
i.e., to the system
V*k = 0 (3.12)
There is an alternative way to describe a solution to (3.12), namely by
replacing the T advections by T-(I-l) new variables defined as closed
circulations of water "loops". The technique is most simply illustrated
by an example. The nine boxes in the upper part of Figure 3 are
connected by 12 advective fluxes and they have no external exchanges.
The 12 numbers inserted show a flow pattern where water continuity is
fulfilled in all boxes. In other words, these 12 numbers would define a
vector m^. It is clear that the same information can also be expressed
by only four numbers using the method in the lower part of the figure.
It is easily realized that every flux pattern, without external gains
and losses, can be reduced from 12 to four numbers in the same way.
In terminology from linear algebra, the observation we just made can be
stated as follows. Since equation (3.12) is homogeneous, its solutions
form a subspace of the space of all vectors m. The dimension of this
space is IFT-(I-I), which is 12- (9-1) =4 in the case of Figure 3. The
four loops chosen constitute a basis in that subspace, i.e., any 31
5 , 3
i 5 2 3
• 13 , 1
1 i 8 15I 4 1 1 8 4
t!)
k*) • i
Ficaire 3. A water-œnserving motion pattern for nine boxes and the representation of the same pattern in terms of "loops". 32 solution HL to (3.12) can be written as linear combinations of loops in a unique way.
When the configuration of boxes is less regular than in Figure 3, the way to establish a set of L loops may be less obvious. However, it can always be done except in degenerate cases (e.g. when the system is composed of decoupled regions) of no interest in the present context. It is important to ascertain that the loop basis is wide enough for all water advection patterns to be represented, since otherwise interesting solutions might not be expressible, and therefore be lost.
The problem of avoiding this is equivalent with the problem of making sure that no loop is a linear combination of the others: if one is able to find L linearly independent loops, these must, by definition, span an
Ir-dimensional space. In a two-dimensional case, such as Figure 3, it is easy to show that the circulations around the L enclosed, finite areas make up a "natural" basis, with the wanted linear independence.
We note from Figure 3 that there will be linear relations between on one hand the T advections m^., t = 1,..., T , that are components of the vector m^, and on the other hand the loops, Y.j, 1 = 1, ..., L, for example,
ml = -y± ' ™3 = ^1
m + *2 = -Y2 ' 4 = Yl Ï2
In matrix form,
n^ = LQ y (3.13)
where LQ is a T x L matrix
Inserting (3.13) in (3.11) we get the following expression for the general solution to the irihomogeneous water continuity equations: 33
m = L-y + m_ (3.14)
3.4 Continuity equations for tracers.
Tracer continuity was dealt with in detail in our previous paper (Bolin et al., 1983). We consider the tracer distribution q ? , where n=l,... ,N identifies the N tracers being considered.
For salinity and the biochemical tracers q^ is expressed as the amount of tracer material per unit mass of sea water. Because of adiabatic heating of sea water when pressure is increased, the continuity equation for internal energy is somewhat more complicated and will be dealt with later. Slightly modifying the derivation given by Bolin et al., (1983) with regard to the units of q we have Ç "(J KM- - ? Kg J "g j* - i (3.15)
where the summations are extended over the values j that refer to adjacent boxes, m-^ is the water flux in the direction perpendicular to the surface -A;j between boxes i and j ; KJJ is the average eddy diffusivity across this surface; FQ^ is the withdrawal of carbon from box i due to new primary production or release from net decay, and F^ is similarily the withdrawal or release of carbon due to new carbonate formation or dissolution; ^0 and %u are the Redfield ratios for tracer n n=l,... ,N as related to these two processes, respectively; Q- is the total amount of the nth tracer in box i; A-\ " I. S the radioactive decay rate for the nth tracer, and finally F^ is the source or sink for the nth tracer at the external boundary for box i. 34
We note that the eddy diffusivity term in our formulation supposedly
accounts for transfer due to motion on scales less than those directly
captured by the advective flux described by the first term in equation
(3.15). Kji accordingly may be considered as transfer by subgrid scale motions and transient (seasonal) flow patterns. In the case of heat
flux, it should be recognized that temperatures will change as a result
of vertical displacement due to adiabatic compression or expansion.
Since compressibility of sea water depends on both temperature and
salinity, pressure changes affect temperature differently for different
salinities, but an approximate treatment can be accomplished by the use
of potential temperature obtained by considering the compressibility of
standard sea water. In the upper parts of the oceans, the spatial
variations of temperature are comparatively large and the pressure
correction plays a rather insignificant role. In the deeper layers,
where small temperature changes due to the compressibility may be of
importance also salinity variations are small and the use of potential
temperature accounts approximately for this effect.
As before we evaluate the integrals in (3.15) by employing a finite
difference formulation:
^:J (3.16) jiWM^jy^
J
where A,-; is the distance between center points of adjacent boxes. It
should be noted that this formulation of centered differences in our
formulation of the advective exchanges might be inappropriate (see
Fiadeiro and Veronis, 1977). As seen from (3.15) and (3.16), the
transfer is partly recipient-controlled. This implies that the time
integral of aj1 in (3.15) might be unstable when we use the given tracer 35 field as the initial condition to derive the changes of this field as dependent on a solution of ocean circulation and biochemical processes as derived with the present method.
When adopting the loop formulation for the advective velocities, the tracer continuity equations are modified. We notice from equations (3.15) and (3.16) that the advective flux of tracer n into box i is given by
1 •f ^ ÏL-f{j "- Z*5j -y" " (3.17) where the summation is over all adjacent boxes j. We replace m— by a set of loops y-v which enter the box i from j and leave it for some box k. The flux of tracer material transported into box i (i.e. f^k) due to loop y.jjç, is given by
fjk= y frjk (qj-*Ji)-(qi+<3k >= Y vjk fy-%) <3-18)
Since the flow of m^ is nondivergent all terms of the kind given by (3.17) can be transferred into expressions as given by (3.18). In other words,
•3 \ A ^k ^j~ ^k) (3.19)
It is important to note here that the replacement of the advective fluxes m^- by loops y-^ means that the average concentrations '/# * ( Hi + i\ ) "that enter as coefficients in equation (3.15) are changed to one half of the concentration difference,
Further we assume (Bolin et al., 1983) that there is no horizontal transfer (from one region to another) of organic and inorganic detritial matter, i.e. whatever is formed in the surface layers of the sea is decomposed below. There is no net accumulation at the bottom of the sea.
This provides 2*R (=24) additional continuity equations
>_ 1- = 0 for all r = 1,2..R (3.20) s=i and
<$• H ^U -O for all r = 1,2...R (3.21) s-,
Here, as for water continuity, it is desirable that these equations are satisfied exactly, which can be achieved by a variable substitution (see section 11).
3.5 The set of diagnostic equations and the inequality constraints
From our conceptual point of view geostrophy has been the basis for the development thus far but from a computational viewpoint water continuity is fundamental. We organize our matrix and the discussion in the next two sections to reflect this latter view. Specifically, we combine the equations (3.14) for water continuity, (3.7) for geostrqphy, (3.15) for tracer continuity (including heat), and (3.20 and 3.21) for detrital continuity and obtain a system of linear equations that will be denoted
A -x = b (3.22) 37
Ma» TurbuUnct Organic tnarganfe Vector Boundary fUixts varlabUi rfttrUus vatlablt* ol conditions unknowns Internat (220) (220) (96) (961 decay Evaporation precipitation A0 0 0 0 »d adjacent exchange (Section Geostrophy- 1 prescribed artas )* 0 0 0 velocity Mass changes fluxes Average Salinity 1220) Adjacent salinities gradients 0 0 exchanges
Turbulence Average Temperature variables Air-sea ond temperatures gradients 0 0 (220) adjacent exchanges
Average Concentration Organic Air-sea and concentrations gradients I I detritus adjacent (96) exchanges
Average Concentration Isotope ratio Isotope ratio Air-sea ond concentrations gradients Inorganic adjacent «0*1) IT?!) exchanges detritus and decoy (96) Average Alkalinity Adjacent alkalinities gradients 0 2l exchanges
Concentration Average Redfield rat'» Adjacent concentrations gradients (Kfl) 0 exchanges
Average Concentration Redfield ratio Air-sea and concentrations gradients 0 adjacent exchanges
0 0 1 1
l^fJiL Arrangeaient of the matrix A, the vector of unknowns x the right hand side vector b of equation (3.22). unKnowns x and 38
where A is a matrix that can be determined from hydrographie data, data
on steady-state tracer distributions and the Redfield ratios. Figure 4
shows the organization of the matrix. The column vector of unknowns,
is given by x = { (mt , t=1,... ,T) , (KfcAt/A t , t=1,... ,T) ,
(F . , i=1,...,l) , (F . , i=1,...,I) )T (3'23)
We note that this system is based on advections (m^.) and not on loops.
The transformations as discussed in section 3.4 are dealt with in
section 5. The "forcing" of our system, b, is the vector of
inhomogeneous terms in equations (3.14), (3.7) and (3.15) and they are
in (3.14) externally imposed water flow defined by bi"1
in (3.7) specified change of geostrophic flow between layers
as determined by Ms /-A., - M^, /-^s+( in (3.15) externally imposed flow of tracers and, in the case
of C, radioactive decay.
To this set we add a set of inequality constraints
G-x > h (3.24)
For the moment, this inequality refers to two kinds of constraints.
a) Turbulent constraints. In the definition of the coefficients of eddy
diffusivity (itrplicit in (3.15); see further Bolin et al., 1983) we
generally wish to impose the condition that
Ky > 0 (3.25)
We have noted, however, that the present formulation of advective
transfer using partly recipient-control in the finite element
formulation may lead to unstable sets of equations for time integration. 39
It is formally possible to avoid this by making -use of the stabilizing property of the eddy diffusivities K^. The condition
K X |/Vn;J ' (3.26) J 06 will ensure stable time integration; however, it is stronger than necessary (sufficient but not necessary) and there is no physical justification for imposing such a condition. The problem of how to best formulate the time integration is further discussed in section 3.6. b) Decomposition of new primary production and the dissolution of carbonate formation. We need to ascertain by using inequality constraints that
-new primary production and carbonate formation only
take place in boxes reached by sunlight,
-decomposition and dissolution in a box in a region can
never exceed the production that has occured in the surface layer
above minus decomposition and dissolution in layers above that of
concern.
These biological constraints are always applied. They are,.however, somewhat more complex than might now appear because we consider regions with more than one surface box (cf Part III).
4 Analysis of matrix characteristics and principles for solution
4.1 Basic characteristics of the system
Our purpose for this section is to clarify the characteristics of the common base that our approach shares with traditional inverse methods for the analysis of oceanic data that have been developed by others 40
(particularly Wunsch, 1978; Fiadeiro and Veronis, 1982; Wunsch and
Minster, 1982; Wunsch, 1984, Fiadeiro and Veronis, 1984; and Wunsch,
1985), who generally have used inverse methods in an indeterminate
(underdetermined) setting. We wish to elaborate further on the reasons
•why we introduce an incompatible (or overdetermined) inverse framework and establish the characteristics of that framework. In the end, we hope that this way of building an inverse structure, piece by piece, will remove some of the confusion about the application of this methodology in constructing models of oceanic processes.
Consider first a purely advective abiotic ocean box-model as defined in
3.1 that uses I boxes, arranged in a network of R regions within which there are vertical flows between S layers and horizontal flows between R regions at each layer (cf Figure 1).
Suppose we seek to determine the water flow, i.e. the T^j^v different terms, from the imposition of the 1-1 independent water
continuity constraints alone. Alternatively one simply replaces the
systems of T advective fluxes with the 1-1 continuity constraints by the
system of L loops, noting that D=Th-+Tv- (1-1) =T- (1-1).
We now add the geostrophic constraints where we assume for the moment
that they can be applied at each of the S layers in each of the W walls.
There are thus W*S additional constraints on the T advections, but we
need to introduce W unknown reference velocities; alternatively, there
are W* (S-l) =Tn-W constraints on just the T advective fluxes (cf. section 3.2). Accordingly the system of L unknown loops satisfying the W*(S-1) geostrophic constraints has an indetermancy of
L-W* (S-l) ^-HEy- (1-1) - (Th-W) =TV+W- (1-1) =R* (S-l) +W- (R*S-1) =W+1-R.
In the particular case we choose in section 3.1, we find that the
resulting indetermancy for computing the advective field is 41 six-dimensional (17+1-12=6), which corresponds precisely to the requirement of determining additionally six loops (cf Figure 5) to define uniquely the two-dimensional flow in one horizontal layer. This, in turn, is equivalent to the necessity of defining a set of reference velocities in order to determine completely the flow by using the conditions of geostrophy and water continuity. There are W reference velocities that need to be determined, and water continuity for each region gives R-l independent requirements. We note that by prescribing a set of reference level velocities or by applying a minimization procedure (e.g. pseudo-inverse; see Wunsch, 1978; Fiadeiro and Veronis,
1982), we can derive a solution that satisfies water continuity and the geostrophic condition exactly.
It is interesting in this context also to recall the method to analyze hydrographie data proposed by Stommel and Schott (1977): the p-spiral.
This method is based on the assumption that the geostrophic relation is satisfied exactly and that accordingly the horizontal divergence and the vertical change of vertical velocity can be derived from the north-south component of the current field and the variation of the Coriolis parameter (@). Although the horizontal velocity can be obtained reasonably well in this way, we also know that the large-scale oceanic circulation patterns are maintained by surface winds and thermohaline processes at the ocean surface which impose dynamic forcing on the oceanic system, and basin configuration and bottom topography which constrain possible motions. They all create nongeostrophic components of the flow and are reflected in the field of vertical velocities and presumably in the large scale distribution of the tracers in the sea. It is, therefore, desirable not to impose the geostrophic condition too severely. Also, inaccuracy of hydrographie data is another reason for not requiring that geostrophy be exactly satisfied. Finally the 42
1
r - -
2 3 ,
k 5
••
6 - 7
8 9
r ni
10 11 r
12
Figure 5. Saras model as in Figurs 1, viewed "from above". If mass fluxes must be exactly geostrophical, the vertically integrated flux has only one degree of freedom at each of the W (=17) walls. Mass continuity for the R (=12) regions gives R-l independent constraints for the W free parameters. Œhe total number of free parameters that govern the mass fluxes thus is W - (R-l) =6
Œhe six free parameters can be represented by the rates of the vertically integrated circulations around each of the six loops. 43 indeterminacy will also be larger than W+l-R, i.e. six-dimensional for the particular system that we have adopted above for illustration because the geostrophic condition in an actual application must not be applied close to the equator.
We return to our example abstract ocean basin model as described in section 3.1. If we also make use of hydrographie information by imposing the continuity relations for salinity and enthalpy (temperature), we obtain 2*1 (=192) additional equations, and the system would shift from being indeterminate to being incompatible. In this setting, however, considering only purely advective solutions is inadequate, since turbulent transfer of salt and enthalpy is essential (cf Wunsch and
Minster, 1982). Including the Tjj+T^T (=220) turbulent transfers as unknown eddy diffusivities shifts the system back to being indeterminate with an indeterminacy in the illustrative case chosen of at least 34 dimensions when applying the geostrophic condition in each vertical surface between boxes. Thus hydrographie data, the condition of water continuity and of geostrophy are, in principle, inadequate for the diagnosis of the ocean circulation, although significant features can be derived. Our system remains iMeterminate.
At this stage another of our objectives (section 1) in building an ocean model becomes important. We are interested in the biochemical interplay in the ocean. In particular, we ask the question: How might we extract the information on oceanic transfer and circulation that is reflected in the global distribution of important biochemical tracers? More specifically, we are interested in questions of the kind: What are the rates of transfer of 002 between the atmosphere and various oceanic regions? Probing such questions implies a need to determine the rate and distribution of various biochemical processes such as primary production and decomposition. As already mentioned we need to employ tracers which 44 carry information about these processes. We use five additional tracers: total dissolved inorganic carbon, (DIC), carbon-14, alkalinity, oxygen and phosphorous, and determine two additional processes for each box: 1) the rate of new primary production of organic tissue and its decay in deeper layers and 2) the rate of new carbonate formation and its decomposition.
In the present case, these additional unknowns give a total of J =
2*T+2*I (2*220+2*96=632) unknowns. The additional constraints raise the dimension of the forcing space (the right hand side) to Ju^yf"
W+(I-1)+N*I+2*R, (136-17+95+7*96+2*12=910), where N (=7), is the total number of tracers. The N*I equations are the continuity equations for the N tracers; the 2*R equations arise from detrital continuity. Thus, the system we have defined is of dimension 910x632.
The preceding account describes a system that is similar to the one we eventually consider. In the remainder of Part I of this paper we continue to use this 96 box model to address formally four additional issues; 1) Some equations, particularly water continuity, should be satisfied exactly (cf. section 3.1 and 3.3, particularly the discussion of loops) 2) The bias implicit in the choice of units, both for the unknowns and for the equations, must be removed. 3) We need to impose a set of inequality constraints upon the solution (cf Section 3.5). 4) The noise in the data set, and hence in the matrix A and the forcing b, must be considered in determining the solution x.
4.2 Equality constraints
The question of which equations need be satisfied exactly and which ones only approximately might appear at first glance to be only an issue for the incompatible systems. The same issue can, however, arise in studying an indeterminate system, either if the system itself is incompatible or 45 if the imposition of constraints poses an incompatibility with the set of exact solutions. In fact, if the dimension of the kernel (null space) of the matrix is small in comparison to the number of inequality constraints, then the system often will not have an exact solution that meets all the inequality constraints (see Bolin et al., 1983). One can approach the problem of satisfying some of the equations exactly simply by weighting heavily those equations (rows), but this can raise numerical difficulties and can be in conflict with the wish to have as
"unweighted" a system as is possible (as we discuss next). The loop formulation (section 3.3) is an approach that is particularly advantageous to satisfying the water continuity equations exactly because it also removes one of the particular weighting issues. Also, the approach is quite general.
4.3 Weighting and scaling
Weighting is a central issue in inverse methodology (Wunsch and Minster,
1982; Bolin et al., 1983; Wunsch, 1985): The metric in the forcing space is obviously important because of its fundamental role in the itdnimization. The need for weighting originates from realizing that the use of ad hoc units and dimensions when formulating the linear equations leads to a metric with undesirable properties. In simple cases, the equations can be made comparable by a change in units, which is equivalent to a multiplication from the left with a diagonal
"equilibration matrix" Dg, i.e. the original system Äx=b is replaced by
The problem becomes more involved when different equations have not only different units, but also different dimensions. In the present case, most equations are in units of mass divided by time. The exceptions are the heat balance equations, which are in enthalpy divided by time; the alkalinity equations, which are in equivalents divided by time; and the 46 thermal-^wind equations, which are in length divided by time.Furthermore,
"mass" is not always mass of the same tracer. We describe in section 12 how we have resolved this dilemma, but we stress that there is no unique or objective way of constructing the equilibration matrix Dg. As a matter of fact D-, must not necessarily be a diagonal matrix. For example, all tracers have their profiles influenced by the patterns of precipitation and evaporation. One may maintain that before any other tracer than salt is incorporated into the system, the effect of evaporation and precipitation on its distribution should be eliminated, in order not to include the same information more than once. Such an operation is equivalent to subtracting a certain multiple of the salt equation from the tracer balance equation, or, in other words, using an equilibration matrix Dg that contains nonzero elements off the diagonal.
Most frequently, weighting does not only aim at removing implicit and irrelevant properties from the metric in forcing space. Normally, one will feel more confident about some of the equations than about others, and therefore upweights these latter equations by multiplying them with some appropriate number. Weighting is thus a "two-step" procedure:
Removing implicit weights and introducing explicit weights. We return to the question of explicit weights in connection with the discussion of results.
While the importance of weighting is obvious in connection with overdetermined systems, the significance of scaling (i.e., the choice of units for the unknowns) is less self-evident. As a matter of fact, scaling is not always necessary. Scaling is essential if and only if one needs to use the notion of distance in the solution space X- For example, scaling is important if one wants to formalize when two solutions J^ and Xg are "similar", or when a solution is "of minimum norm". In the present study, there are two reasons why scaling needs to be considered. Firstly, in sections 15 and 16 we compare solutions in 47 terms of the correlation and regression between them. It is easy to realize that these coefficients are dependent on the definition of distance in X. Secondly, in order to search for "almost underdetermined" aspects of the equation system, we perform a singular value decomposition (section 15.4). This technique involves a base of unit vectors V in X. The definition of a "unit" vector obviously presupposes a measure of "length" in X.
There is no all-purpose method for scaling. As stated by Golub and van
loan (1983, p. 74):
"The basic reaDmmendation is that the scaling of equations and
unknowns must proceed on a problem-by-problem basis. General scaling
strategies are unreliable. It is best to scale (if at all) on the
basis of what the source problem proclaims about the significance of
each a^ •• Measurement units and data errors may have to be
considered."
In section 12, we describe different methods for scaling, applicable to the present problem.
4.4 Inequality constraints and noise in the data
In our previous paper (Bolin et al., 1983), we required the solution to meet only a set of half-plane constraints. This simplifies considerably the numerical treatment (Moore and Björkström, 1986). However, we now recognize the importance of more complex constraints. The issue arises,
for example, if we use the formulation of the finite element expression for the advective term in the tracer continuity equation (cf. sections
3.4 and 3.5 eq (3.26). In this formulation, as we have already pointed out the flow of a tracer between two boxes is partly recipient-controlled. This is important because recipient-control can lead to undamped oscillations in the concentration of a tracer when the solution is time-integrated in the study of the transient behaviour of a 48 tracer with time-varying boundary conditions (eg. C02 investigations). We also encounter this difficulty in the determination of the tracer distributions that would satisfy the given system exactly, to be compared with the original tracer distributions used in the inverse calculation. We consider the complexities in more detail later (see further Part IV). For the moment, it is enough to realize that simple sign-constraints for K5.1 may no longer be adequate. Hence a matricial formulation for the inequality constraints is generally necessary.
In the consideration of noise in the data, one must recognize that there are a variety of avenues to approach this difficulty. Many may, however, not yield to computer implementation. Our concept is to search for an optimal solution by an iterative two stage minimization process. First we seek the best fit solution for a given data set and then consider the possibility of improving the "fit", through alterations of the data, within acceptable limits, i.e., alterations of both the matrix A and the forcing vector b. This will be described in more detail later in section
17.4.
5 Derivation of a solution
We consider formally the issues of imposing conditions which demand that: a) the water continuity equations are exactly satisfied; b) the linear system is unbiased by implicit weights, and c) the system is solved under convex constraints.
In part, what follows is but a careful restatement of the construction in the previous section. We begin again with a discussion of the water continuity alone rather than starting with geostrophy.
The water continuity equations can be expressed in terms of advections as an IxT (i.e. 96x220) system (AQ, tP), where ^ is the matrix that arises from the linear expression for the continuity of water for each 49 box and b_ represents forcing. Because of the requirement of overall mass balance, the system is rank deficient and one equation (row) can be eliminated; (see section 3.3). We can think of AQ as a linear transformation from the space M of unknown advections to the space E^ of known water forcings (cf Figure 6a). In this context, there is a natural linear identification of the T-I+l (=125) dimensional vector space Y of closed loops for this system of I (=96) boxes and T (=220) advections with an important subspace of M. Namely, one can linearly identify (cf section 3.3) the space of loops with the null space (K) or kernel, K=AQ(0), of AQ (here and elsewhere in this paper the superscript
+ connotes the pseudo—inverse). Denote this identification by Le: Y-?K in M. Also in the T (=220) dimensional vector space of advections, M, there is a T-I+l (=125) dimensional hyperplane S of exact solutions to the indeterminate system Aç^t™; in other words, S is the set of advections which satisfy exactly the water continuity under a given forcing represented by b™. One of these solutions is a particularly useful building block, namely n^a^fcP1, being the solution with minimum norm.
The larger system that includes the other unknowns (i.e., the eddy diffusivities, the rates of new primary production, decay and decomposition) and the additional equations (geostrophic flow, tracer and detritial continuity) may be considered as a simple extension of the system L4 : Y -*M and AQ : M ^Sm- Specifically, we first attach the T (=220) turbulent unknowns, the I (=96) unknown rates of new primary production or decomposition, and the I (=96) unknown rates of new carbonate formation or dissolution to both the space of loops, Y, and the space of advections, M. The resulting J^IrfT+2-I (=537) and Jv (=632) dimensional vector spaces are denoted by Z and X, respectively.
Correspondingly, the space, JL, of forcings, is expanded in terms of the figura 6a. Diagram depicting linear transformations for solving (3.10) involving the linear spaces £, M and B^, and the linear operators 51
Figure 6b. Expansion of the space Tf to space 2t, space JJ to space JJ, and space B_ to space B. Space g is further transformed to an unweighted space Sn Plf tne operator Dg. Œhe space IJ of inequality constraints is also indicated, together wxth the operator G from 5J to H. Ao
Un ro
Ficaire 6c. Extension of the diagram in a) to illustrate the transformations required to solve (5.2) including inequality constraints and unweighting. ïhe uppermost part is the same as in a. In the middle the spaces Y and M have been augmented to Z and X, respectively and B^ has been augmented to B. The lowest part shows the space H, which consists of all right hand sides of the inequalities and the unweighted space B^. 53 normalized thermal wind increment in the geostrophic equations, the boundary conditions for the tracer equations and 2*R (=24) coordinates that are needed to express continuity of the regional detritial continuity. This expanded space of forcings is denoted by B, and it is of dimension Jb=(I-l)+W*(S-l)+N I+2R (=910). The relational information between these additional unknowns and the forcing as defined above is then included by an expansion of the matrices LQ and AQ, yielding an enlarged system Lï Z^X and A: X-»B, see Figure 6b. If we denote by b the actual vector of enlarged forcing, we have re-stated the
incompatible linear system (A,b) that we defined in section 3.5.
Classically, it is in this system, (A, b), that one seeks an x such that jj Ax-b| is minimized over the set of x's which, in addition,
satisfy a given set of convex constraints, such as nonnegative eddy diffusivities. The exact character of these constraints has been
indicated in section 3.5. For the moment we express them abstractly as a given matrix G, a vector h and the relation Gx>h. As stated this setting
4(A,b, G,h)lis, in general, the setting for analysis. However, it is unsatisfactory in two regards. First, we have not yet singled out the water continuity for exact solution and secondly the system is still
implicitly weighted. We have, however, almost all the mathematical
elements required to resolve these difficulties. Figures 6b and c will provide a useful roadmap.
Our wish to satisfy the water continuity equations exactly brings into play the loops; more specifically, the expanded space, Z, (see Figure
6b). In order to use the space, we first need "to move" m.0=A0 iP to the enlarged space X. This is done simply by attaching to the T (=220) dimensional vector m_, the Jv-T (=412) dimensional zero vector. Let x_ denote the resultant vector in X. In addition to the linear matrix 54
L:Z-»X, we need the nonlinear operator N:Z->X defined by the relation
N(z)=L(z)+xQ. Recalling the discussion in section 3.3, we are ready to consider satisfying exactly the water continuity equations in solving the system (A,b) ; for the moment without constraints (G,h). In this
case, one simply solves the incompatible Jj-,xJz (=910x537) dimensional
system (A*L,b-Ax0) ; lets z denote the solution, and sets xs=N(zQ). However, when we are faced with constraints (G,h) on x, we must proceed
in a slightly different manner since the constraints apply to vectors in
X and not in Z. This can be resolved by replacing the system (G,h) of convex constraints on X, by the equivalent system (G*L,h-GxQ) on Z.
There remains now only the issue of unweighting the system, and since this also alters either X or Z as well as B, we are forced to change
again the linear system and the exact expression of constraints. A
change in weights on the dimensions of either Z, X, or B can be regarded
as the multiplication by a diagonal matrix. We postpone discussing the
actual convex constraints that we employ until a later section, and we
also delay prescribing the exact formulation of the reweighting
matrices. For the moment consider them as invertible (no diagonal term
zero) diagonal matrices
Dz : Z-^, Dx : X^ : Dg : B-^ (5.1)
where we are using the subscript, u, to denote the spaces that are
unweighted.
We have shown above how we can satisfy the water continuity equations
exactly in (A,b) and fulfil the constraints (G,h) by moving to the
"loop" space; therefore, we need only unweight this environment and
reform accordingly the systems (A/b) and (G,h). Again, Figure 6c should
serve as a useful roadmap to the transformation. 55
We begin with { (A,b), (G,h)j and assume that Dz, Dx and Dg are given. (We
shall find below that the actual realization of Dx is not necessary.
This will prove fortunate, since determining Dx could be troublesome.)
Also assume that L: Z->X is as before, that xQ is known and hence N is defined.
To solve in an unweighted environment the constrained minimization
{(A,b), (G,h)jone would solve the system { (DB*A*DJ£,DBb), (G*Dx,h)|. But this would leave the water continuity equations only approximately
satisfied. Therefore, rather than work with this system we combine this
procedure with the preceding discussion of the nonlinear operator N and
linear matrix L and transfer to an unweighted loop environment: i.e. the
spaces Zu and E^. Specifically, the system (cf. Figure 6b and 6c) of interest is
{(Dg A L Dz, DJJ (b-Ax0)), (G L D2, h-GxQ) j (5.2)
A constrained solution minimizing errors can now be determined for the
incompatible system defined by (5.2). We denote this solution by z^ and —1 . . further define xs by xs=N(D2 z^). We claim xs satisfies a) the continuity equations for water, b) it minimizes the unweighted problem,
and c) satisfies the system of constraints (G,h).
Firstly, xs satisfies the water continuity equations exactly since it is
in the image of N, or putting it another way, xs can be written as
xs=xQ+X[C + Xj» where the advections in xQ satisfy exactly the water
continuity; the advections in xfc are in K, the kernel or null of AQ, and hence have no role in the water continuity as expressed in either A or
A; and the advective terms in the remaining vector Xj. are all zero.
Secondly and thirdly, x was determined in a manner that removes any
implicit weights in the solutions and it minimizes the unweighted 56
problem (Dg A, Dg b) and satisfies Göoh. This can be seen by observing
first that, by construction, ^ resides in an unweighted space and hence
the units of measure for the unknowns have been removed from
consideration, and we have set xs=N (Dgi^)• That xs satisfies the unweighted minimization, and the constraints can be seen from two
calculations.
-lDsLA(NCï);2a))-Ui-ilD8(Axs-4)ll
and
dLD^z^K-Cx,^ ClNU£'aJ)>^ ^ Cx5 >h (5.4)
By these transformations our problem can be solved by using an existing
program for matrix inversion least squares minimization under convex
constraints (cf. Haskel and Hanson, 1981.) 57
PART II. THE Dftlft. BASIS
6 Overview
We have chosen to apply the inverse method as developed in previous sections to deduce the large-scale circulation and biochemistry of the
Atlantic Ocean because here the data required are more plentiful.
It is important to choose the model structure carefully as we are restricted to use a rather limited number of boxes. Coastal currents play a crucial role in ocean dynamics. It certainly would be desirable to consider their role in the present context. There are, however, a number of studies which have rather focussed on the meridional structure of the flow in the Atlantic Ocean with prime emphasis on the thermohaline circulation leaving out the explicit consideration of coastal currents. The resolution adopted here does not permit an adequate description of both these aspects of the problem and we do not consider explicitly the role of boundary currents in the present formulation. Their role for the transfer processes will implicitly be accounted for by the processes of turbulent exchange.
The following sections present in some detail the data fields, i.e. the internal geostrophic constraints, the tracer distributions within the domain chosen and the external conditions in terms of exchange of water, enthalpy and tracer material with the atmosphere and adjacent seas. As will be seen, however, we face considerable difficulties in formulating the appropriate boundary conditions which express the exchange of water and tracers between the Atlantic domain and the Pacific and Indian
Oceans. Since these fluxes are not well-known our solutions may not describe accurately the flow in the Atlantic to the extent it does depend on an interplay with these other oceans. On the other hand, this 58 offers, as we shall see, an opportunity to stud/ the sensitivity of our solution to variations of boundary conditions. As a matter of fact the analysis of the sensitivity of the solution to the choice of model, to the formulation of the physical-chemical and biological processes which have been included and to the uncertainty of the data form a very central part of the later discussion and is fundamental in trying to judge the usefulness of inverse methods in the interpretation of tracer data.
7 Box topology
The domain of the Atlantic Ocean which we consider is shown in Figure 7.
We acknowledge the importance of the Mid-Atlantic Ridge for the deep sea circulation by dividing this domain into a western and eastern basin.
Regions 1 and 12 include the Arctic Sea and Antarctic waters, respectively, the latter in east west exchange with the Pacific and
Indian Oceans. (In most of the experiments that we report on we have not considered exchange between region 11 and the Indian Ocean, which certainly takes place in reality. We will analyze the sensitivity of this assumption.) These two regions are considered as boundary reservoirs and the part of our solution that refers to processes within these regions can not be considered as reliable because of the difficulties in formulating accurate boundary conditions. The ten interior regions are defined on the basis of well-known oceanic features. Regions 2, 3, 10 and 11 are in the latitudes of westerly atmospheric flow associated with surface currents towards the east.
Regions 4, 5, 8 and 9 include the subtropical gyres and 6 and 7 cover the tropics. The boundary between regions 9 and 11 is placed along the
Walfish Ridge, acknowledging its role in closing direct communication in the deepest layer between the eastern Atlantic basin and Antarctic waters. 59
^
^ S ^
•*-Ti^v ï
8
10 11
12
Figure 7a. The definition of regions 1 - 12 as used in the analysis of the Atlantic Ocean. 60
LEGEND + = GEOSECSI x =TTO
Figure 7b. The same definition as in Figure 7a. The distribution of GEOSECS and TTO observations is also shown. 61
Figure 7 shews that the Caribbean Sea and the Mexican Gulf have not been included, but the water flow through these ocean basins from region 6 to region 4 is instead prescribed as a boundary condition (cf section
8.3.1). Similarly we permit, inflow of water from the Arctic Sea to region 2 through Davis Strait (cf Aagard and Greisman, 1975) and exchange of water between region 5 and the Mediterranean Sea.
We next turn to the vertical division of our domain (cf Figure %.
Simple descriptions of the abyssal Atlantic usually identify three water masses:
1) Antarctic Bottom Water, which penetrates northward from the
Antarctic region into the western basin below the crest of the Mid'
Atlantic Ridge. Water is supplied to the eastern basin through the
Romanche fracture zone close to the equator. Based on a section of linear relation between depth and latitude, sloping from 3000 m at 60 °S to 4200 m at 80 °N. The Antarctic Bottom Water occupies the lowest of our model's eight vertical layers. 2) North Atlantic Deep Water which has, on the average, larger vertical extent than the Antarctic Bottom Water. We therefore represent this water mass by two layers, 6 and 7. The dividing line is at a depth of 2500 m, which was chosen after inspection of a section showing alkalinity corrected to 35% salinity and also for the influence of nitrate released in decomposition of organic material. The scalar thus obtained reflects the signal that carries information about the dissolution of CaCO, (which is an important group of unknowns in our problem). The lower part of the North Atlantic Deep Water is different from the upper part in this regard. 62 WESTERN SECTION LEVELS Figure 8. Vertical N-S cross-section along the set of GEOSECS observations in the western basin as shown in Figure!.:. The density surfaces shown are chosen as boundaries between layers 1-6. Two planes separate the layers 6-8 (see further text). 63 3) Antarctic Intermediate Water, is formed as water sinks in the subpolar convergence zone in regions 10, 11 and 12. We let this water occupy layers 4 and 5 (see Table 1). Northward of the equatorial region the layers 4 and 5 describe the lower thermocline region primarily being supplied from the Gulf Stream region. Depth is used in defining boundaries between the deepest layer, whereas 0"&' surfaces define the boundaries between the uppermost six layers, (cf. Table 1) This is done because tfd is not a relevant vertical coordinate in the deep oceans (Lynn and Reid 1968) and depth proved most convenient for our purposes. Rather using 5^ or & surfaces for the deeper boundaries would not alter the geometry significantly. In region 1, however, the dividing level between layers 7 and 8 was chosen so as to give the two layers equal volume. The choice of 01 surfaces is a compromise between getting evenly spaced boundaries and an attempt to resolve features of Antarctic Intermediate Water. This water mass is a clear feature of all tracer fields and deserves attention on its own right. The differences between the particular fields we have chosen can be resolved by the current model. This is of interest in view of the fact that Antarctic processes are inadequately treated in most carbon cycle models developed to date. The uppermost warm surface water and the thermocline region at lower latitudes ( Table 1. Vertical structure of layer configuration. ( f denotes latitude). Separating layer surface - 0^25.80 - (T, - 26.60 - 01-27.20 Ol- 27.40 6- 07-27.60 & ï- 2.500 m 2- 3.000+1.200 (f+60)/140 (-60° é. f ^ 80°) 65 Given the topology as now defined, it is straightforward to count the number of surfaces, at which one component each for advective and turbulent transfers needs to be defined as described in section 2. Water can exchange either vertically between the adjacent layers in the same region or horizontally between two neighbouring regions in the same layer. There are, however, some restrictions in the latter case. The Antarctic Bottom Water is too shallow to reach above the crest of the Mid-Atlantic ridge. No east-west exchange can therefore take place in layer 8, except through the Romanche Fracture Zone between regions 6 and 7. The Walfish Ridge in the same way prevents flow between regions 9 and 11 in layer 8 and finally, the sill between Greenland, Iceland and Scotland excludes layers 7 and 8 in region 1 from direct aartmunication towards the south. On the other hand, there are six cases where region borders at the surface coincide with 6*& -isolines that separate the layers (cf. Figure 9). Therefore, for example, surface water (layer 1) that moves horizontally southward out of region 8 passes directly from region 8, layer 1 to region 10, layer 2. In addition to the transfer within layers or vertically between layers there thus arise six such additional "slant" connections. In the end, the total number of components describing advective and turbulent transfer is 368, i.e. 184 for each kind. 8 Data 8.1 Geostrophy The division into regions as described in section 7 introduces 17 vertical boundaries between the 12 regions. The number of layers present at a boundary, denoted by S in section 3, varies between five and eight. There are thus S unknowns that denote flow of water between any two 66 Figure 9. Map shewing the areas vftiere the layers 2-8 reach the ocean surface. The regions 1-12 are shown as a background. 67 regions, and we use the thermal^wind equation (3.7) to formulate S-l linear relations between these. The data necessary to compute the areas A and the fluxes Mg in (3.7) are taken from the compilation of temperature and salinity measurements by Levitus (1982). This data base has a resolution of one degree latitude by one degree longitude and resolves 33 layers in the vertical, at specified depths below the ocean surface. The boundary between regions 1 and 2 is considered too narrow to permit reliable data extraction from Levitus's compilation, and the boundary between regions 6 and 7 also has to be left out, since the thermal wind equation is hardly applicable close to the equator. For each of the remaining 15 boundaries, between four and seven relations are derived. The total number of equations thus formed is 80. The coefficients in these equations are derived using the following technique, suggested by Ulf Cederlöf. We describe any state variable on a boundary by a set of data points, separated horizontally by roughly 100 km and in the vertical defined by the 33 vertical layers. In this way a two-dimensional grid is defined in a yz-plane, where y denotes horizontal distance along the boundary. Using algorithms given by Millero et al. (1980) for density as a function of temperature, salinity and pressure, and by Fofonoff (1980) and Bryden (1973) for potential temperature, we obtain in situ densities ? (yn, zn) and potential densities G"ö (yn, zn) at all the grid points (yn, zn). The S areas SLS can be calculated by scanning all the grid points and computing, for each one, an area element A, which is part of an -A^ surface or split between two such surfaces, depending on the values for z_ or The mass flux Mg as defined by (3.4) and (3.6) may be written and is obtained by a similar procedure. We first evaluate the function £(Yn, V 3 ^v^v,) "- J IM (8-2) 5--0 This integral is calculated by summation of terms for all i = 1, ... (n - 1). The y-derivative of b is then approximated by the difference formula whereafter, the surface integration in (8.1) can be carried out by scanning all the gridpoints similarly to when computing^ The areas A. s, and the mass fluxes Mg relative to the surface are given in Appendix A, Table A3. The ratio M^-A*. has dimension velocity (if implicitly assuming a constant water density of 1000 kg/m3). It illustrates the change of the flow from level to level if the geostrophic balance were exactly fulfilled. 8.2 Tracer distributions Through the GEOSECS (Geochemical Ocean Sections Study) and TTO (Transient Tracers in the Ocean) programs a rich data set is available. We attempt to make use of all the data. Extracting information from the nutrient tracers is particularly central in this work. The tracers available from all samples in GEOSECS and TTO are salinity, temperature, 69 dissolved inorganic carbon, alkalinity, phosphate, dissolved oxygen, nitrate and silicate. Nitrate is not used because of its strong similarity to phosphate, which means that little independent information can be extracted from the nitrate distribution. Including nitrate would only introduce an almost redundant set of equations. The use of silicate is also avoided because it would introduce a new set of unknowns, i.e. the opaline detritus. Since the opaline detritus rain cannot be determined by the other tracers, silicate would not increase the determinacy of the model. Although alkalinity likewise introduces a set of unknowns (the CaC03 detritus) the scientific interest in 002 demands a careful study of the alkalinity and carbon cycles. The remaining tracers are all used and presumed to contain independent information. All tracers (except N03 and K>4 discussed above) differ from the rest by at least one physical process (cf. Table 2). In order to utilize both GEOSECS and TTO data corrections must be applied to some of the tracers to achieve mutual consistency- The corrections adopted are a decrease of DIC and alkalinity from GEOSECS by 14 micromol/kg and 14 microeq./kg respectively as discussed in the literature. (Takahashi et al., 1985; Broecker and Takahashi, 1978; Bradshaw et al., 1981). Figure 7 shows the GEOSECS and TTO stations in relation to our regionalization. Although the model originally was tuned to utilize the GEOSECS data a paucity of data in some regions is obvious. Region 3 was particularly a problem until the TTO data became available. In hindsight, now having the TTO data set, one might adopt a finer grid scale for regions 1-5. In particular, the large and heterogeneous region 4 may be criticized. On the other hand, the lack of data from the interior of the southern gyre region remains a severe inadequacy for regions 8-11. Nevertheless we utilize the availability of two data sets with high quality analyses to explore the model 70 Table 2. Overview of how different processes being considered in this study affect the distribution of tracers and enthalphy (temperature). Note that not all these tracers were eventually included in our system of equations, and that enthalpy, although formally included, was downweighted by 0.001 in the reference case. Tracers 14 Processes Sal. Temp. DIC C Alk. P04 02 N03 S^ Advective transp. xxxxxxxxx Turbulent transp. xxxxxxxxx Organic detritus X X X X X X Ca003 detritus XXX Opaline detritus X Atm. heat exchange X Attn, water exchange XXXXXX XXX Atm. gas exchange XX X Radioactive decay X 71 sensitivity to differences in the cx>ncentration fields that are used in the confutation. With the aid of TTO data we also identify model inadequacies due to the structure of the GEOSECS data. A more extensive discussion about representativity of data in space and time is given in section 16, based on comparisons of GEOSECS and TTO data. The procedure for acquiring tracer concentration values for each box is simply to form arithmetic averages of all samples available from each domain. The values extracted in this way are given in Appendix A, Tables A4 and A5 both using GEOSECS and the combined GEOSECS and TTO data sets. The determination of concentration distributions for DIC and C finally deserves some special comments. Because of increasing atmospheric concentrations enhanced transfer to the oceans of both 002 and C has taken place. Present concentrations in the sea therefore do not describe steady states. lacking better information we still adopt the values for DIC as observed during GEOSECS and TTO expeditions. It is possible to determine a posteriori the implications of such an assumption. The changes of C, since bomb-produced C was first injected into the atmosphere in 1952, have been much larger in relative terms. We therefore try to assign values valid as of 1957, at which time bomb-produced injection still were small and for which year about 200 measurements are available, (Broecker et al. 1960). For a more detailed discussion cf. Broecker and Peng (1982) and Bolin (1986). 72 8.3 Boundary conditions 8.3.1 Water Precipitation, evaporation and river flow Baumgartner and Reichel (1975) have compiled available data on precipitation and evaporation over the globe. We use their maps for the Atlantic Ocean to evaluate for each region the flow of water between the atmosphere and the sea as shown in Figure 10. The adopted values are given in Appendix A, Table A6. The spatial distribution over the ocean is, however, not accurately known, particularly not for the South Atlantic. As we shall see later the solution of our matrix equation is markedly dependant on the boundary conditions that define the vector b. Indirectly the tracer equations are also influenced in that precipitation and evaporation diminish respectively enhance tracer concentrations in the surface reservoirs. This is implicit in the model formulation in that the components of the b-vector for the different sets of tracer equations depend on b^ (cf. section 11). We explore in section 16 the sensitivity of our solution to uncertainties in the water flux boundary conditions. We note that the total net evaporation of water from the Atlantic Ocean is 27-1012 tons per year (ta-1), i.e. 0.87 Sverdrup. This corresponds to 35 mm per year for an area of the size of the Atlantic. There is a significant inflow of fresh water to the Atlantic Ocean by rivers. Based on data given by Kempe (1982) we give their contributions in Appendix A, Table A6. In addition there is some seepage of ground water in the coastal areas between these major rivers. A comparison of the areal extent of the drainage basins of the rivers to the areas that 73 Figure 10. Net transfer of water (cm a~x) between the atmosphere and the sea due to precipitation and evaporation based on the analyses by Baumgartner and Reichel (1975). Ihe values adopted for the regions are shown within boxes. ïhe dashed lines indicate where the interfaces between layers reach the sea surface (cf. Figure 9 >) whereby the surface of the regions are divided into subrogions as considered in the model. 74 do not feed into these shows that this additional contribution cannot be very significant. Water exchange with adjacent seas Inflow from the Arctic Sea takes place through the Davis Strait and we assume that 63-1012 ta-1 (= 2 Sv), as given by Aagard and Greisman (1975), enter layer 3 in region 2. Surface water in the Arctic Sea is diluted by the fresh water from the Siberian rivers and a rather intense current enters region 1 through the Fram Straits between Greenland and Spitsbergen. Due to inadequate tracer data from this region of our model, we consider this whole region as a boundary region to the rest of the model and merely include a net fresh water influx that is equal to the water inflow to the Polar Sea due to the Siberian rivers. There is also water exchange between the Atlantic Ocean and the Mediterranean Sea through the Straits of Gibraltar. According to Sverdrup et al. (1942), 55«1012 ta"1 (1.75 Sv) of surface water leaves the Atlantic while 53-10 ta"1 (1.68 Sv) of more saline and colder water is returned to deeper layers. This dense water sinks and mixes into the Atlantic deep water and is very obvious from hydrographie data. We do not know well how it is distributed vertically and assume tentatively the dicharges as given in Appendix A, Table A7. The westerly, wind driven ocean current in the equatorial belt that reaches South America turns northwest-wards along the coast, enters the Caribbean Sea and the Mexican Gulf and feeds the Gulf Stream through the Florida Straits. The intensity of this major current varies with season. The average flow through the Florida Straits has been given by Stommel (1960) based on data from Montgomery (1941) to be 880 • 1012 ta-1 (= 28 75 Sv). We adopt this value for the amount of water entering region 4 and assume that it is distributed in the vertical in accordance with Montgomery's data (see Appendix A, Table A8). The data from Baumgartner and Reichel (1975) show net evaporation from the Caribbean Sea and the Gulf of Mexico (cf. Figuref» ). Some compensa tion is due to the outflow of the Mississippi river. For the time being we assume that the outflow from region 6 into the Caribbean is the same as the return flow to region 4. Although we also assume that there is neither any changes of the water properties, this by-pass is important since the direct flow between regions 6 and 4 is controlled by our assumption of quasi-geostrophic flow. As a matter of fact this boundary condition will influence our solution markedly. We note that the specifications given above yield a total net inflow to the Atlantic Ocean of 43*10 ta , which water must be exported by the Antarctic Circumpolar current. The water transport through the Drake Passage (into region 12 of our model) has been estimated by several authors. We shall adopt a value of 3800 • 1012 t a-1 (120 Sv). The net outflow towards the Indian Ocean is of course equal to the sum of this inflow and the net gain referred to above. Obviously data are insufficient to determine the boundary conditions to regions 12 in order to describe properly the influence of region 12 on transfer processes within the Atlantic Ocean. We therefore merely assume that the net outflow (43 -1012 ta-1) is distributed in the vertical over the five layers (4-8) in region 12 in a way that is as similar as possible to the distribution of water inflow to the Atlantic as described above. 76 8.3.2 Exchange of tracer material and enthalpy with adjacent seas Basic assumption Our basic assumption is that the system we study is in a steady state. Consequently, we must prescribe inflows and outflows in such a way that their overall sum is zero for each tracer. For carbon, radiocarbon, oxygen and enthalpy, the balance involves atmospheric exchange terms. These are estimated in section 8.3.3. For radiocarbon, the balance also includes radioactive decay. Of all the oceanic and atmospheric fluxes that enter the picture, the least-known ones are the water-borne flows from and to the Pacific and Indian Oceans. We therefore estimate those fluxes last, assigning to each layer a value simply obtained by the overall balance requirement. The gradients between region 12 and its surroundings that we then implicitly assume, are small, since the water flow here is 3800 • 10 ta (120 Sv). Thxs procedure may be considered as a first approximation and is difficult to improve, since data are inadequate. Exchange with the Mediterranean, Arctic and Caribbean Seas We assign values for the inflow and outflow of enthalpy and tracer material from and to the Mediterranean Sea as follows: The net exchange of salt is assumed to be zero and the salinity of the return flow from the Mediterranean to the Atlantic is therefore enhanced by a factor of 55/53=1.038 as compared with the Atlantic water entering the Mediterranean Sea (cf. section 8.3.1). We apply the same principle for phosphorus. Further, Mediterranean water is enriched in dissolved inorganic carbon and alkalinity. The concentration values for the water returning to the Atlantic is determined from the GBOSECS observations in 77 the western parts of the Mediterranean. Data for oxygen concentration and temperature are similarly available. There are good reasons to assume that the C content of the top layers of the Mediterranean Sea, with which we are concerned, was in close equilibrium with the atmosphere before bombtesting and we accordingly assign the value A 14c= -50 o/oo to the return flow from the Mediterranean into deeper layers of the Atlantic. The values adopted are given in Appendix A, Table A9. The supply of enthalpy and tracer material from the Arctic Sea through the Davis Strait is assessed on the basis of data given by Aagard and Greisman (1975), while the exchange with region 1 and the rest of the Arctic Sea is ignored. The flow into the subtropical gyre from the Mexican Gulf is assumed to have the same temperature and tracer concentrations as the water in the western tropical Atlantic (Region 6), from where this water originates. 8.3.3 Air-sea exchange of heat, carbon dioxide, radiocarbon and oxygen Finally we consider the air-sea boundary conditions that are needed to apply the conditions of heat enthalpy tracer balance as described in the previous section (8.3.2). Enthalpy The exchange of heat between the atmosphere and the ocean has been studied by many authors, because of its importance for the global climate. For our purpose the work by Bunker (1980) and Esbensen and Kushmir (1981) has been particularly useful. We use a map given by the former author to estimate box integrals for the heat fluxes for each box that reaches the ocean surface. Our estimates are given in Appendix A, Table A10. 78 The heat equations basically express conservation of enthalpy. In order to obtain correct expressions for the gains and losses of enthalpy in the surface boxes, it should be observed that water that evaporates or precipitates brings its enthalpy content with it (as if the water flux had been to an adjacent ocean rather than to the atmosphere). We have included this effect on the right-hand sides in the heat equations, based on data for net water exchange between the atmosphere and the sea as shown in Figure 10 and given in Appendix A, Table A7. We have, thus, implicitly assumed that the evaporating water, as well as the precipitation, has the same temperature as the box with which the exchange takes place. Carbon and radiocarbon Air-sea exchange of carbon dioxide is only approximately known. Data are still insufficient for an accurate mapping of the carbon-dioxide partial pressure of surface water and marked variations occur in the course of the year. The dependence of the transfer rate on winds is not either well-known. We assume that the net flux of carbon from the atmosphere to a surface reservoir can be written as the difference between two gross fluxes, Fa^ - F^a, that are given by the kinetic equations t'A. where Ai is the area and ?i the partial pressure of C02 in reservoir i. We determine the two proportionality constants, For carbon, we have s > " '^7"> S (8.4) where the symbol X stands for summation over the surface boxes, E is export to adjacent oceans and G is accumulation. Since we use fractionation-corrected data for radiocarbon, we can write *n«JA ~~ ^W* + E*+ 6*+ *>* •- ° (8-5> where Rj_ denotes the ratio of 14C atoms to carbon atoms. The decay term •k D can easily be computed from the adopted values for R^ and total carbon concentration in the sea. We base our analysis on the steady state distribution of tracers in the sea, which means that E = G = E =G =0. It should be noted, however, that the concentration of dissolved inorganic carbon has been influenced by the invasion of carbon from the atmosphere due to the increasing concentrations of carbon dioxide in the atmosphere and therefore does not represent a steady state distribution accurately. This is similarly the case for radiocarbon, although the AC values for sea water as used have been based on an approximate pre-bomb distribution (cf section 8.2). Presumably there is therefore in reality a net downward transfer of DIC and C into the ocean. Furthermore there is, in all likelihood, some export (E > 0; E > 0) at the present time, since the global oceanic uptake of C02 probably is more intense in the Atlantic region, from where the carbon is transported to the Pacific and Indian Oceans through the Antarctic circumpolar current. However, we do not know the size of E or E , nor do we know if they were zero xn a pre-rndustrial steady state. The global carbon cycle may have involved net outgassing 80 of C02 from the Pacific/Indian Ocean, balanced by uptake in the Atlantic. The export term for radiocarbon is subject to similar uncertainties. In our reference case solution, we set all four numbers E, E , G and G equal to zero. By solving (8.3) and (8.4) for k and X we deduce an annual exchange rate of 13.6 mol/m , which is somewhat smaller than most other estimates. It should be pointed out, however, that a value for the exchange rate computed indirectly in this way much depends on explicit or implicit assumptions made in course of the deduction (cf. section 16.7). The air-sea fluxes of carbon and radiocarbon as deduced are given in section 16 (Table 10). Oxygen We are not able to prescribe air-sea exchange of oxygen due to lack of adequate information. We therefore assume that this transfer is zero and accordingly no exchanges with adjacent seas either take place. This assumption is not very satisfactory and accordingly all equations for oxygen balance in surface reservoirs will be down-weighted when we solve our basic system of equations. 81 PART III REALIZATION OF TOE MODEL FOR THE ATLANTIC OCEAN 9 Overview Part III of this report is essentially technical. It is included largely for the benefit of those who wish to check the results, or want to penetrate carefully into the details of the construction of the matrices and vectors that occur in equation (5.2). Section 10 describes how we have applied the method described in section 3.3 to ascertain water continuity. Section 11 illustrates the organization of the matrix and also describes formally how to obtain equality between biologic production and decomposition in each region ("detritus continuity"). Section 13 is an application of the principle for deriving a solution as described abstractly in section 5, and contains also a detailed statement of the inequality constraints we have used. Except for the definitions of our terms, "row unweighting" and "block unweighting" (section 12), it is not necessary to read Part III in order to appreciate the presentation of our results, which follows in Part IV. 10 The loops: Construction of the matrix L and the vector mQ We next describe how to apply the "loop" technique as introduced in section 3.3 to the configuration of our model as designed in section 7. Since we use 84 boxes with 184 interconnecting surfaces, the number of loops is T-I+l=184-(84-1)=101 (cf. section 3.3). However, the loops must not be drawn arbitrarily. Care must be taken so that none of them becomes a linear combination of the others. 82 A logical way to divide the present model is to consider the eastern and western halves separately. We choose (arbitrarily) to count the polar regions with the western part. We then get two two-dimensional fields and apply the loop seie^ion method to each of these, as sketched in Figure 3, section 3.2. The western half consists of 47 boxes and 80 advections, and thus gives rise to 80-(47-1)=34 loops. In the eastern half, 37 boxes are joined by 62 advections which define another 26 loops. There are, finally, 42 advections that connect the two planes, and it is easy to realize that this calls for 41 more loops. When defining these, one can proceed quite arbitrarily, but must be careful to follow the rule, that each new loop must go through some surface between boxes, where none of the loops already defined has gone. In this way we ascertain that none of the loops can be a linear combination of loops previously introduced. It follows from the arguments in section 3.3 that we have thus found a satisfactory way of representing all those patterns of water advection that are consistent with the given external exchanges. In other words, we have constructed a matrix LQ as defined in equation 3.13. (Each of the loops corresponds to a column vector in LQ). The element with smallest norm, m0, can now be found either by explicit construction of the pseudo-inverse AQ and computing AQ b™ or by another procedure that does not require construction of A* and is therefore favourable for computer memory reasons: It is easy to find some HL in M for which AQ BL equals the given b™. The element mQ can then be found by applying a projection operator on HL: *o = (X-Vb) "p f10'1) where I is an 184x184 identity matrix and LQ denotes the pseudo-inverse of L0. 83 Having thus set up LQ and m0, we carry out the variable transformation from m to y, where nt=I*+m0 as in equation 3.11. 11 Construction of the matrix A and the vector b. Let the 536-dimensional vector x be composed of the following elements, (as ordered below). 184 mass fluxes, "advections", m^, as defined by equation (3.9), in units of 1015 ta-1. 184 eddy diffusivities, "turbulences" i.e. expressions Kj.•<&..//i .. -LJ 1J J-J as defined in equation (3.15) and (3.23) in units of 1015 ta'1. 84 organic production terms, one for each box, as defined in (3.15), in units of 10 moles of carbon a . 84 calcium carbonate production terms as defined in (3.15), in units of 1015 moles of carbon a . We can then write the 84 continuity equations for tracer n in matrix form: j> A* feZ Yh ]•* = £ (11.1) where A is an 84x184 matrix such that A^t indicates the coefficient for m^. in the tracer n continuity equation for box i. As seen from (3.16), A^t=l/2* (q^+gjj ) if xtu. connects box i with box j, where j < i. -Aj£=l/2*(q*|+q.!j) if m^. connects box i with box j, where j > i (sign convention). 84 ci A^t=0 otherwise A is analogous to Ä , but contains instead the coefficients for the turbulent terms, i.e. from (3.16): 4 - qj - qj if |4|- V2* I is the 84 x 84 identity matrix, and jfc and jfc are Redfield ratios as defined in (3.15) bn is an 84-dimensional vector where the ith component n n bf = - A"Qi + Fbi (equation 3.15). <*• T ... We combine the two matrices A and A to an (84x382) dimensional matrix A11L^- j where the superscript n indicates that tracer n was used in the construction. Here, n=C,*,A,P,0,S, or T. We combine water continuity and all the tracer continuity equations and the geostrophical equations to one system: A x = b (11.2) 85 where A and b are composed of blocks as follows: *b 0 0 b 0 0 bG I I b T T b* 0 21 b* A = I 0 and b= bp I 0 0 o i rf»i 0 0 J ."Ti The diagonal matrix T appearing in the radiocarbon equations is derived from the assumption that there is no isotopic fractionation in the biological processes. When one unit of carbon dissolves in box i, we thus simultaneously get a supply of T^ units of radiocarbon, where T^ is the ratio of 14C atoms to C atoms in the top box of the region concerned. The slight fractionation in the biological process is insignificant in the present context. The matrix A has 80 rows, one for each of the thermal-wind equations (3.7) that can be formulated in the model. There are two nonzero elements on each row of A, which correspond to the coefficients and - The right hand sides of (3.7) form the 80-dimensional vector b , to be evaluated from temperature and salinity data. We next replace x by z where the 184 advections have been replaced by 101 loops, using equation (3.14) L„ o" n* x= i/ • z+ = L-Z+X„ (11.3) 0 1 .° i 86 The matrix L implicitly defined by (11.3) corresponds to the operator L : Z-*X described in section 5. Thus, equation (11.3) in fact defines the non-linear operator N: Z-»X mentioned in that section. The least-square minimization problem becomes A > (L z + x0) Z b which is equivalent to A L Z &b-A x0 (H.4) In setting up the matrix A and the vector b, the detritus continuity equations are left out. Exact satisfaction of these equations is ascertained in the following way: The unknowns that denote organic and inorganic production in the top box of each region (eq. 3.23) are replaced by the negative sum of the corresponding terms for the other boxes in the same region. It is easy to see that this manipulation is equivalent to a matrix operation whereby z is replaced by z'= E z, where z' has 24 components fewer than z and E defines the substitution just described. Similarly, the operator L is replaced by L* = E L. These steps are not illustrated explicitly in Figures 6a-c or in equation (11.4), but they can be thought of as simple modifications on the space Z and the operator L. Since water and detritus continuity will automatically be fulfilled, the corresponding rows can be omitted from the matrix A«L. We illustrate equation (11.4) schematically in Figure 11, which shows the matrix A»L, the vector z and the right-hand side b-A X . This Advective Turbulent Organic Inorganic Vector Boundary variables variables detritus detritus of conditions. "loops" variables variables unknowns Internal (»101) («184} (=72) (=72) decay Thermal wind (Section Geostrophy- 1 equations areas)* 0 0 0 Advective prescribed velocity (»80) modified by LQ variables changes "loops" Adjacent S (101) Gradients Gradients 0 0 basin (=84) exchanges Turbulent Air-sea and T variables adjacent Gradients Gradients 0 0 (=84) (184) basin exchanges Air-sea and OIC adjacent Gradients Gradients 1 1 Organic (=84) basin detritus exchanges oo variables Air-sea and WC (72) adjacent Gradients Gradients Isotope ratio Isotope ratio (=84) exchanges To V* and decay Inorganic Adjacent A detritus Gradients Gradients 0 2 basin (=84) variables (72) exchanges P Adjacent Gradients Gradients Redfietd ratio 0 basin (=84) exchanges Air-sea and O adjacent Gradients Gradients Redfield ratio 0 (=84) basin Y'o° exchanges AL b-Ax o Figure 11. Arrargeroent of the matrix A L, the vector of unkna the ritfit hand side vector b-ÄxQ of equation (11.4). 88 equation system forms the natural starting point for a discussion of 1) how to remove any inadvertent weighting, caused by the use of different or ill-tuned units for different equations, and 2) how to formulate the inequality constraints (section 3.5) on the components of the vector Z. 12 The iinweicthting; Construction of matrices Dz and Dg Since different equations are expressed in different units (and have different dimensions), and since the units for the unknowns are also different, the elements of the matrix A*L, see (11.4) may differ markedly in magnitude. Ihereby, errors in the equations will affect the least-squares computation differently, as discussed in section 4.3. This is an unintentional weighting that should be eliminated, before a solution to the system is sought. Consequently, we have to modify the matrix A' L so as to obtain a matrix whose elements are as close to unity as possible. There is no objective way to accomplish this. One may pick any matrix Dg and by definition say that in Dg-A-L-ZSfDg (b-A X ), all equations have the same weight. To illuminate the non-uniqueness of Du,, we describe two different procedures for the equilibration, which we term row unweightim and block unweightim, respectively. In row unweightingy we divide each nonzero element in the matrix by the Euclidean length of the row vector it belongs to. Thus, after the row weighting, the equality H (Dg A L)^ = 1 j holds for all i, which we find to be one reasonable way to specify that all equations have the same weight. 89 Block unweiqhtincr is a more complex process, in Which more attention is paid to the physical interpretation of the equations. To get an overview of the matrix A-L we divide the equations into five groups and the unknowns into four groups. The matrix A-L then consists of 20 blocks, as shown in the upper part of Table 3. We then proceed as follows: 1. The unit for loops and turbulences is changed so that the absolute values of the (nonzero) elements in block Rj have an average of unity. (This operation affects blocks B^, B^ B4 and B5 also). 2. The weight on the enthalpy (T) equations is changed so that the elements in block B2 get unity average. The same procedure is applied to the salinity equations (S). 3. The weight on the alkalinity equations (Alk) is changed so that the elements in block B^ get unity average. (This operation affects block Bg also). 4. The unit for carbonate production is changed so that the elements of block Bg get unity average. (This operation affects block Bg also). 5. The oxygen (0) and phosphorus (P) equations are divided by their respective Redfield ratios, so that all organic production terms get the absolute value unity. (This operation affects block B5 and 6. The weight on the geostrophic equations is changed so that the elements of block B-j^ get unity average. This completes the description of the block unweighting procedure. 90 Table 3. Block structure of the matrix A. The symbols B.^ express the magnitude of the coefficients in the subsystems of the matrix, i.e. "the blocks". The values in the lower part indicate the order of magnitude of the coefficients in respective blocks after block unweighting. Matrix Loops Turbulences Organic Carbonate production production Geostrophy Bl 0 0 0 S and T B2 B2 0 o DIC and C14 *3 B3 B6(=l) B8(=l) ALK B 4 B4 0 Bg(=2) P and 0 B5 B5 By 0 Unweighted matrix Geostrophy 10 0 0 S and T 11 0 0 DIC and C14 1 1 B6(=l) B^g/BjBg AIK 11 0 1 P and 0 Bj/B^B, BVB^By 1 0 91 After these steps, most of the block elements are centered around unity as the lower part of Table 3 shows. The exceptions are: - The coefficients for loops and turbulences in the phosphorus and oxygen equations. The magnitude of a coefficient here is a characteristic P (or 0) gradient, divided by the product of a characteristic carbon gradient (B1) and the Redfield ratio (By). In reality these numbers are also close to unity. The coefficients for carbonate production in the total carbon equations. The magnitude of a coefficient here, BQ being unity, is the ratio of a characteristic alkalinity gradient to twice (Bg=2) a characteristic carbon gradient. These numbers on average are of the order 0.1-0.2, which can be seen as a reflection of the relatively minor role that carbonate production plays for the balance of dissolved inorganic carbon. It is easy to see that steps 2, 3, 5 and 6 are equivalent to a shift from the problem (11.4) to a problem that can be written Dg A L Z % Dg (b-A X0) (12.1) while steps 1 and 4 are equivalent to a variable substitution z = Dz Zu <12*2) where Dg and D^ are diagonal matrices. We thus get the formulation DEALDzZu^bu (12.3) where bu = Dg (b-A-X0) 92 Block urweighting is thus not only a multiplication with a diagonal matrix from the left (i.e., "weighting" in the usual sense), it also involves unit changes for some of the unknowns (i.e., scaling, a matrix multiplication from the right). However, although unit changes will give us a different set of numbers as the solution to our problem, the physical meaning of the solution is not affected thereby. 13 The inequality constraints; construction of a matrix G and vector h It is convenient to begin this section by considering the vector x as introduced in section 11, since the inequality constraints that we wish to impose are most directly expressible in terms of the components of x. For instance, the conditions of nonnegative turbulence (3.25) give xi > O for i = 185, ..., 368 (13.1) and the requirement that primary production and carbonate formation must not go on in boxes not reached by sunlight also means that x± > O (13.2) for certain i-values that can easily be identified. The third kind of inequality constraint we impose deals with the boxes that are partly sunlit and partly covered by other boxes. We can not a priori constrain the sign of the bioproduction here, but we can require that dissolution, if it goes on, must not proceed at a faster rate than is permitted by the export of detritus from the boxes above the box in question. To take a specific example, suppose four boxes i, j, k, 1 are located on top of each other, and suppose that i, j and k reach the surface, while 1 does not. We can then require for the bioproductions X.! f x_; / XT_ cinci x*i • 93 Xi < O X^ + X^ + Xk < O (13.3) x^ + x^ < O x-j^ > O Inequalities such as the second one are not simply sign constraints, but it can be transformed to the form II G^- Xi > 0, which has already been considered. Finally, constraints are also imposed on a few of the advective unknowns. The advection through the Romanche Fracture Zone is required to be eastward and not smaller than 160-lO12 t a-1 (Schlitzer, 1984). Size constraints are also posed on the Greenland-Iceland flow. In summary all the inequality constraints are formulated as G x > hQ (13.4) •where the elements of G are +1, 0 or -1 in an obvious manner. The number of rows of the matrix G is around 330. The variable transformations (11.3) and (12.1) give G (LD^ + Xj >h0 (13.5) which is equivalent to G L Dz 2u > ho " G Xo <13'6> In the framework of Figure 6c, we view the vector hQ as an element in a vector space H. The matrix G corresponds to an operator G: X H. There is a certain subset H in H where h>h0-Gx0 is fulfilled, and a certain subset x'inX such that if x is in x' Gx is in H. Continuing the chain, there is a subset Z1 in Z such that if z is in Z1 then L-z is in X , and, finally there is a subset z'u in Zu such that if z^ is in Z , then D~ • 94 ^ is in Z1. The product operator G1 = GID^ goes from Z i to H and it is easy to prove that G ^ > h - GxQ if and only if Gx > hQ. The operator G is indicated in Figure 6c. Summarizing (12.3) and (13.6) we see that we have two operators defined on Z^ °EÄLDz : V*Ai G1 : ^-»S We have also identified two points, b^ and hQ - GxQ such that h^ is the point we want to approximate, and Gz^ > hQ - Gx0 are the side conditions we need to satisfy. This is exactly the form under which the equation system is solved. The algorithm we use has the name "Subroutine ISEI" and has been described by Haskell and Hanson (1981). A solution requires 20 seconds CTU-time on a Cray-1 computer. 95 Bart IV RESULTS 14 Overview The results of the experiments so far conducted are of two kinds. 1) We have analyzed the solution vector x with regard to its physical- chemical- biological characteristics, with consideration of the uncertainties of the model, the method of solution and the data used (sections 15 and 16). 2) We have learned about the intricacies and difficulties which are encountered when using matrix inversion methods and some of these may be of general interest for workers in the field (sections 16 and 17). The system with which we are concerned is overdetermined. It is shown that the subset of enthalpy equations is incompatible with all other tracer equations. This probably is due to the fact that temperature is the decisive parameter that determines the density distribution and accordingly the static stability and vertical mixing. Furthermore marked seasonal variations of the temperature occur in the surface layers and the mean annual temperature distribution is therefore not the appropriate variable for determination of turbulent heat flux. Enthalpy has accordingly not been included as a tracer in most of the experiments that have been made. The role of weighting is illustrated by systematic variations of the weight on the set of geostrophic equations. Both a small weight (0.1) and a large one (50) create features of the solution which qualitatively seem unrealistic. On the basis of an analysis of these experiments a weight of 4 is chosen for most experiments in the later analysis. 96 A careful analysis of a reference solution is given. We recognize major features in the pattern of advective flow, which are due to the enforcement of the condition of ouasi-gecstrophic flow. A large scale meridional circulation with northward flow in the surface layers and southward flow in the deep sea with an intensity of about 470 »10 tons per year (ta-1) (15 Sv) is derived. The field of turbulence is less well defined. The mean value of the turbulent water exchange between adjacent boxes becomes less than half of the mean advective flow from one box to its neighbour. An analysis of the rate of detrital matter formation, decomposition and dissolution is given. Although the detailed pattern probably is guite uncertain the values for the yearly total new primary production in the Atlantic (2.2 1015 g organic and 0.3 1015 g inorganic C) are in reasonable agreement with results from other studies. Section 16 explores the model's sensitivity to some basic model assumptions. Uncertainties in the data that are used to produce the coefficients of A, and finally variations of the model assumptions are studied. The boundary condition experiments are limited to studying the influence of water, 002 and C. Uncertainties in the coefficient matrix are explored in two ways. By creating a data-set only utilizing the GEOSECS results and comparing solutions obtained then with solutions obtained when also including TTO/NAS and TTO/TAS data some uncertainties are found. By successively rounding off the tracer data to fewer decimal places some numerical aspects of the model results are determined. 97 Basic model assumptions are studied by varying the "Redf ield Ratios" in a set of experiments and by producing purely advective solutions (i.e. all turbulences are set to zero). Although the sensitivity studies could be extended in various ways the results from these experiments are sufficient to draw conclusions about the utility of this model for 002 uptake studies. Conclusions about the methodology of applying inverse methods to a tracer field to extract highly resolved information about ocean circulation with a model of this type also emerges. Section 16 concludes with a discussion on reasons for the encountered sensitivities and possible means to overcome some of the difficulties. Fundamental to our methodological discussion in section 17 is the concept of incompatibility. When computing a solution, we inevitably face the question how to act in front of a set of contradictory information. This question was commented on in general terms in section 4.3. In section 17 we return to the subject, now using the present model as an illustration. We discuss the two principal origins of incompatibility, i.e. inaccurate data and approximate equations. We describe some ways that we have explored in order to reduce the effects of incompatibility. We touch upon well-known techniques in matrix calculation, like total least squares and separable least squares, and describe why these methods, at present, do not meet our requirements. We also describe an iterative, approximate method to deduce a solution to the problem of total least squares. Finally, we try to reduce the sensitivity to noise in the data by making a singular value decomposition of the matrix, and setting the smallest singular values equal to zero. 98 15 A set, of basic experiments 15.1 Some introductory remarks Before presenting a reference solution in section 15.3 it is important to discuss some general characteristics of our solution. We first note that the solution to our equation Ax b of course is x = O if b = O. The nonzero components of b that "force" our model to a solution x + O are of three different kinds: 1) Prescribed velocity differences between adjacent layers as given by the thermal wind equation (3.7) 2) Radioactive decay of14C 3) Fluxes of water, tracer material and enthalpy across external boundaries. This includes both the direct transfer of tracer material and enthalpy as given in sections 8.3.2 and 8.3.3, and the "virtual" transfer brought about by the requirement of exact fulfilment of the water continuity equation, which affects all tracers in all boxes as described in section 8.3.1. These three factors provide "the clock" to the system. An inspection of the components b^ shows that b is almost exclusively determined by the condition of geostrophy and the flux boundary conditions, i.e. radioactive decay of 14C contributes comparatively little because of the long half-life of C (cf. section 16). Our boundary conditions are quite uncertain while hydrographie data permit a reasonably reliable determination of the geostrophic condition. On the other hand we do not know how stringently it should be applied. 99 We presented in section 12 methods for unweighting our set of equations in order to give the different tracers and the geostrophic condition equal weight in determining a solution. The difference between applying "block" unweighting and "row" unweighting will not influence the results of the following analysis. We choose to use "row" unweighting, which gives the equations for the deep layers a more pronounced influence on the solution, than is the case for "block" unweighting. Our system is greatly overdetermined. The vector x has 429 independent components, while there are 668 equations and 337 inequality constraints. The least square solution that we derive is far from being an exact solution. We define the total error \;=( J £l 1*1 J-' J and the contribution from a subset of equations (referring to a particular set of tracer equations or the geostrophic equations) is denoted by in 1 Ut o where i^ is the row number of the first equation in the set. By successively applying a large weight (50) to the subset of equations referring to the tracers or geostrophy we derive eight solutions Xj^ that are particularly tuned to these subsets of equations. We compare these new solutions with the unweighted reference solution x by 100 cairputing the correlation and the regression coefficients. The results are displayed in Table 4, columns 2 and 3. The correlations are reasonably high for all solutions except in the case when temperature is upweighted (r=0.24). Accordingly temperature is hardly considered at all in deriving the reference solution, i.e. the equations for conservation of enthalpy are quite incompatible with the rest of the equations. We then recall that the thermohaline circulation as well as vertical turbulence are controlled by the density distribution, which in turn primarily depends on the temperature distribution. Furthermore the temperature in the surface layers and part of the thermocline region have a marked seasonal variation which affects the static stability significantly. Experiments show further that the prime reason for this incompatibility of the enthalpy equations is that the boundary fluxes (b^) seem to be ill-tuned to the internal field of temperature. One may therefore question the approach followed here to consider enthalpy as a tracer that can be considered in the same way as the other passive tracers. The use of salinity may perhaps be questioned in an analogous way. Since, however, the correlation coefficients for the solution obtained with upweighted salinity equations is of the same magnitude as for the solutions with upweighting of other tracers, we will retain the set of salinity equations, but we will not make use of enthalpy as a tracer in the following experiments. Having in this way modified our basic system, new solutions and the corresponding correlations and regression coefficients have been derived, cf. Table 4, columns 4 and 5. We note that the correlations with the new reference solution were markedly increased for solutions with upweighted tracer equations C, C, A and S and only slightly decreased in case of P and O. The solution with upweighted geostrophy on the other hand agreed less well in this case. We may view this in the following Table 4. Correlations and regression coefficients between the sol uti ons^^ i.e. the subsets of referring to the different tracers '*i and geostrophy have been upweighted by a factor 50) andX^(the reference case with no upweighting). Columns 2 and 3 refer to the solutions using the complete set of tracers, while columns 4 and 5 show the results when enthalpy ( T ) is not considered. Columns 6 and 7 show the corresponding results if comparison is made with a weight of 4 for the equations of geostrophy. Complete set of equations Enthalpy eq. not included Geostrophic weight, 4 Enthalpy eq. not included Subset of Correl. Regression Correlation Regression Correlat ;ion Regression equations between coefficient between * coefficient between coefficient upweighted between A^x andX,t between X^ and X^ between by 50 and X,i andX^ X^and Kt C 0.753 0.865 0.921 0.855 0.740 0.517 14n 0.594 0.735 0.912 0.773 0.774 0.494 A 0.803 0.980 0.958 0.983 0.789 0.609 P 0.959 0.958 0.951 0.896 0.740 0.523 0 0.968 0.947 0.952 0.898 0.743 0.527 S 0.787 1.58 0.824 1.14 0.566 0.586 T 0.239 0.351 - - - - Geostrophy 0.738 0.897 0.650 1.12 0.918 1.19 102 way. Having eliminated the incompatible enthalpy equations the general similarity between the tracer equations (particularly C, 14C, A, P and O- r between 0.91 and 0.96), imply that they dominate the reference solution and the geostrophic equations are in some important aspects in discord with these tracer equations. This conclusion is further supported if comparison is made with a reference case in which the geostrophic equations have been given the weight 4 instead of unity, cf. Table 4, columns 6 and 7. The similarity between the reference case and the solution with upweighted (50) geostrophy then naturally is enhanced (r increases from 0.65 to 0.92), while the similarity with solutions with one or the other set of tracer equations upweighted all decrease (r between 0.57 and 0.79). We also find that the regression coefficient is considerably less than unity for the solutions with successively upweighted tracer equations (0.49 to 0.61) in contrast to the solution obtained when geostrophy is upweighted which yields a regression coefficient well above unity (1.19). This means that the rate of ocean circulation and new primary production (and accordingly detritus flux) are increased when the condition of geostrophy is more strongly imposed. In the next section we shall further consider the implications of the geostrophic condition. 15.2 The quasi-cfeostrophic condition We next analyze how the subsets of equations for the different tracers and for geostrophy contribute to the total error in the solution. Table l 5 (column 4) shows the values of (i )n and percentage distribution of errors in the case of unweighted equations. The similarity of the different tracer distributions (except that for salinity) dominate the solution in that the errors for each set are merely between 4% and 13.5%, while that of geostrophy is about 26% and that of salinity 30%. 103 Table 5. The contribution (£)Kfrom different subsets (n) of equations to the total error ^^for different weights on the geostrophic equations (upper table) and their percentage distribution (lower table). The magnitude of the ageostrophic flow compared to the imposed geostrophic constraint is also given. All values are computed by inserting the solution obtained from the set of weighted equations into the unweighted set. Geostrophic weight 0.1 0.5 1.0 4.0 8.0 50.0 C 0.011 0.014 0.021 0.043 0.058 0.076 wc 0.019 0.021 0.029 0.067 0.105 0.155 A 0.013 0.014 0.015 0.036 0.058 0.098 P 0.013 0.015 0.020 0.039 0.056 0.080 0 0.002 0.004 0.009 0.027 0.035 0.049 S 0.024 0.047 0.065 0.122 0.157 0.212 Geo 0.626 0.157 0.057 0.006 0.001 0.000 et-2" 0.707 0.272 0.216 0.340 0.470 0.670 ageo- strophy 1.71 0.857 0.514 0.163 0.074 0.004 C 1.5 5.2 9.7 12.6 12.2 11.3 IHc 2.7 8.0 13.5 19.7 22.4 23.1 A 1.8 4.6 7.0 10.6 12.4 14.6 P 1.8 5.7 9.3 11.6 12.0 11.9 0 0.3 1.5 4.3 8.1 7.4 7.4 S 3.4 16.0 30.0 35.8 33.4 31.7 Geo 88.5 58.9 26.2 1.7 0.2 0.0 104 The total relative error, £ = 0.216 shows that the solution x still does not satisfy the system of equations very well. We next explore how these values change as dependent of weighting of the geostrophic equations. The solution x contains ill horizontal advective flows, while 80 geostrophic constraints are imposed. The remaining 31 degrees of freedom are determined with the aid of the tracer equations which, however, demand nongeostrophic flow to yield an optimum solution. The fact that the geostrophic equations only contain the horizontal water flow means that we are able to satisfy those conditions exclusively to any prescribed degree by upweighting these equations. We have successively applied the weights 0.1, 0.5, 4, 8 and 50 to the geostrophic equations and have derived the corresponding solutions. The distribution of errors among the different subsets of equations is shown in Table 5. The table also shows how well the condition of geostrophy is being satisfied. In case of no weighting the ageostrophic flow is 51.4% of the imposed geostrophic constraint, but already the application of a weight of 4 has reduced this figure to 16.3%. There is no physical reason to impose a condition of geostrophy more strictly, but it is still of interest to inspect table 5 in some more detail. We first note that i. is a minimum for a weight of about unity for the geostrophic equations. For a large weight (50) the total error ( £ ) becomes large because the requirement of almost exact geostrophy is in conflict with the tracer equations, for I which the errors ( C L accordingly are markedly increased. A small weight on the geostrophic equation, on the other hand, permits the tracer equations to be much better satisfied, while the flow becomes strongly ageostrophic. As a matter of fact the ageostrophic flow component on the average is 1.71 times the mean of the prescribed geostrophic components. The errors in the geostrophic equations completely dominate ( C ) becoming 88.5% of the total error £ • 105 The correlations and regressions between all solutions are shown in Table 6. There is hardly any correlation between the solution with a small geostrophic weight (0.1) and those for which the condition has been strongly imposed (weight 4) and we conclude that the general water circulation as deduced in the traditional way bv consideration of quasi-geostrophic flow is not reproduced in an analysis only making use of tracer information. We further notice that the regression coefficients are larger than unity when relating a solution with a large weight to one with a smaller weight. This further supports our previous observation that the intensity of the circulation increases as geostrophy is more strongly imposed. The use of only tracers tends to yield a slow circulation field. The characteristic differences between the solutions that are obtained with different weights for the geostrophic equations can also be seen by direct inspection of the flow fields. For any geostrophic weight the large scale advective flow is towards the north in the upper layers (1-5) and southward in the deep layers (6-8). The intensity of this direct thermohaline circulation varies, however, markedly with the magnitude of the geostrophic weight. Figure 12 shows the magnitude of the southward return flow in the deep layers (6, 7 and 8) as a function of latitude and weighting. The weak circulation for small weights (0.1, 0.5 and 1.0) is obvious as compared with the more intense flow for large weights (4, 8 and 50). We notice, further, that although the flow is quite close to geostrophic balance already for a weight of 8 (cf. Table 5), there is a significant change of the mean meridional circulation when the weight is further increased to 50. Peculiar features of the flow appear, however, for a weight of 8 and particularly 50. A strict application of the condition for geostrophic flow thus causes very strong downwelling at great depth in equatorial regions and some strong 106 Table 6. Correlations and regression coefficients between solutions with different weigths on the condition of geostrophy. 0.1 0.5 1.0 4.0 8.0 50.0 correlation coefficients. 0.1 1.00 0.67 0.40 0.15 0.13 0.13 0.5 0.67 1.00 0.90 0.61 0.52 0.47 1.0 0.40 0.90 1.00 0.82 0.72 0.65 4.0 0.15 0.61 0.82 1.00 0.98 0.92 8.0 0.13 0.52 0.72 0.98 1.00 0.97 50.0 0.13 0.47 0.65 0.92 0.97 1.00 regression coefficients 0.1 1.00 0.90 0.52 0.15 0.11 0.5 1.00 0.86 0.43 0.32 1.0 1.00 4.0 1.09 1.00 8.0 1.10 1.12 1.00 50.0 0.18 0.85 1.122 1.199 1.099 1.00 107 15 10 Sverdrup .-1 ton yr h40 1.0- 0.5- N60 50 40 30 20 10 Equ. 10 20 30 40 50: 1 I ,_! I I , I X ' . ' I L L_ Regions 2 and 3 4 and 5 6 and 7 8 and 9 Figure 12. Southward advective flow in layers 6-8 as a function of latitude and as dependent on weighting of the geostrqphic equations (0.1; 0.5; 1; 4; 8; 50). .: oas» • closed circulation cells, the existence of which seems unlikely. Strong turbulence is similarly encountered in this region. We should note in this context, however, that there may be errors in the hydrographie field on the basis of which the geostrophic condition has been formulated. The importance of this uncertainty has so far not been further explored. Also small weights for the geostrophic equations lead to more intense small scale flow fields, presumably because comparatively minor errors in the data fields may yield such features if a moderate geostrophic control is not imposed. The fact that large weights on the geostrophic equations increase the mean advective flux in the model (and accordingly the rate of overturning) is also reflected in the total flux of detritus from the photic zone into deeper layers. This is shown in Table 7. The choice of weights when solving an overdetermined inverse problem of the kind dealt with here cannot be made purely objectively since we do not know to what extent inadequacy of the model, insufficient resolution or inaccurate data contribute to the incompatibility. It must rather be decided with regard to the purpose of the study. We may choose to focus on satisfying the dynamic constraint of geostrophy, or we may rather wish to derive an internally consistent view of the carbon cycle based on available data. The choice of weights would obviously be different and dependent on our state of knowledge. It was stated in the introduction that a main purpose of the present analysis would be to see how we can use the inverse methodology to unify for ocean circulation studies using hydrographie data and biochemical diagnosis as pursued by chemical oceanographers. Since altogether six tracers (C, C, A, P, 0 and S) refer to the latter problem and merely the quasi-geostrophic 109 Table 7. Total detrital flux from the photic zone, derived for different geostrophic weighting. Geostrophic Detrital weighting flux (unit 1015gCa~1) regions 1-11 0.1 2.77 0.5 2.57 1.0 2.63 4.0 3.01 8.0 3.18 50.0 3.22 110 condition stems from dynamical considerations it seems appropriate in the following more general discussion to accept some increased weight to this latter condition, presumably between 1.0 and 4.0. When presenting the main features of a reference case in the next section we shall apply the weight of 4 to the geostrophic equations. There is obviously a subjective element in this choice. 15.3 A reference solution As justified in the previous section the reference solution to be presented in this section has been obtained by downweighting the equations for conservation of enthalpy by 0.001 and upweighting the geostrophic equations by a factor of 4. Some comparison with the unweighted solution will also be made. An analysis of the sensitivity of the solution to uncertainty of data and assumptions about relevant processes will be given in section 16. Due to the complexity of our model and the size of the vector x, we shall describe its major features in a series of simplified graphs. The complete solution is given in Appendix B. 15.3.1 The field of advective flow We present first an overview of the circulation by displaying the average horizontal field of motion in i surface waters, i.e. layers 1-3 at midlatitudes and in equatorial regions, including also layers 4-5 in polar regions where these approach the ocean surface; water above about 600 m depth is in this way included (Fig 13a). Ficaire 13a. Advective flow (in 1013 t a-1) in the surface layer (layers 1-3 in regions 4-11; layers 4-5 in regions 1-3 and 12) as deduced for the reference case. Flow to and from intermediate or deep waters in the different regions (cf Figures l£b and l^c) is shown by 9 and G respectively. Hie transfer through the Caribbean Sea and the Mexican Gulf, flow to the Mediterranean Sea and from the Amazon river, which have been prescribed as boundary conditions, are also shown. Figure 13b. Horizontal advective flow (in 1013 i a"1) in the intermediate layer (layers 4-5 in regions 4-11) as deduced for the reference case. Œhe exchanges with the surface water and the deep water are shown in Figures $a and 3£b respectively. Figure 13c. Advective flow (in 10iJ •£ a~x) in the deep -waters (layers 6-8) as deduced for the reference case. Flow to and from intermediate surface water is shown by and respectively (cf. Figures 1$A and oHh). Flow through the Romanche Fracture Zone (dashed line 16 101 g a"1) and the flow through the bottom layer 8 in the eastern basin (regions 3, 5, 7 and 9) is shown separately. 114 ii intermediate water, i.e. layers 4-5 at midlatitudes and in equatorial regions (i.e. regions 4-11); water between about 600 and 1300 m depth is included (Fig 13b). iii deep water, i.e. layers 6-8; water nasses below about 1300 m at low and middle latitudes are included; the layers approach the ocean surface in regions 1-3 and 12 (Fig 13c). We notice the following features: The North Atlantic surface currents are controlled by the flow through the Caribbean Sea and the Gulf of Mexico with a prescribed intensity of 810 »10 ta . A major part of this flow continues northward outside the North American east coast (490-10 ta ) and across the Atlantic Ocean IP —i , south of Greenland and Iceland (660-10 ta ). If a smaller geostrophic weight is applied a much weaker anticyclonic circulation is obtained in the North Atlantic and the forcing imposed by the flow through the Caribbean Sea is to a considerable extent compensated for by a direct return flow from region 4 to region 6 (cf. Figure 14a). The northward flow in the surface layer of the southern Atlantic takes place along the African coast. The water crosses the Atlantic south of the equator and continues northward outside the Brazilian coast to feed into the Caribbean Sea. An intense eastward current prevails through regions 10 and 11, caused by the inflow through the Drake Passage south of South America, and leaving the Atlantic into the Indian Sea. Because of the uncertain boundary conditions this current is not accurately determined. The flow of intermediate water in the northern Atlantic is largely a reflection of the flow in the surface layers, but is weaker. The northward flow of intermediate water from the convergence zone in the 115 southern Atlantic is only present in the eastern basin, while the flow is southward in the west. It does not penetrate beyond the equatorial region. A closer analysis of these features in relation to earlier analyses of the intermediate water formation seems warranted. Deep water is formed in all three northernmost regions of our model at a total rate of 470*10 ta-1, i.e. 15 Sverdrup. Strong westward flow from region 3 to region 2 is obtained, which is in qualitative agreement to the penetration of tritium into the western basin of the deep Atlantic (cf. Jenkins and Rhines, 1980) and the flow of chloro- fluorocarbons as shown by Weiss et al., (1985). The southward flow is most intense in the western basin, although some water sinking into the deep layers in region 5 (presumably caused by the inflow of saline and comparatively heavy water from the Mediterranean Sea) also moves southward. In the equatorial region the flow through the Romanche • • • • . i ? Fracture Zone, layer 8, attains the minimum permissible value, 160*10 —i . . . ta , that is given as a constraint. In addition the solution is characterized by eastward flow in layers 6 and 7 of the equatorial region, and the southward flow in the deep sea of the Northern Atlantic thereby shifts from the western to the eastern basin and continues southward in the eastern basin of the southern Atlantic. Layer 8 in the eastern basin is connected horizontally with adjacent areas only through the Romanche Fracture Zone, but water is also descending from layer 7 in the equatorial region. This water spreads southward and northward and rises into layer 7 elsewhere in the basin in a qualitatively similar way as has been derived by Schlitzer (1984). For comparison we show in Figures 14a and 14b the advective flow pattern as obtained from the unweighted equations. The general features of the flow remain the same, but the flow is less intense. We also note, for example, that the flow in the equatorial surface layers change to Figure 14a. The. same as Fig l3a but shewing the solution obtained with unweighted geostrophic equations. Figure 14b. The same as Fig Qc, but shewing the solution obtained with unweighted geostrpphic equations. 118 eastward flow. The sensitivity to perturbations of the data field and boundary conditions will be further discussed in section 16. There is, thus, considerable uncertainty in the numerical values as given. We also notice that small scale circulation patterns appear locally. This occurs where the tracer gradients between adjacent boxes are small and where accordingly rather large advective flow rates only cause moderate fluxes of tracer material. 15.3.2 The field of turbulent exchange In the present study turbulence has been defined as the exchange of water between neighbouring boxes due to (unresolved) motions on scales less than the one defined by the size of these boxes. Based on the eddy diffusivity concept we have further imposed the condition that turbulent transfer of matter is in the direction of the negative gradient, i.e. the eddy dif fusivity coefficient K^O. About 50% of the diffusivities become zero. The root mean square magnitude of the rate of turbulent exchange of water between adjacent boxes are given in Table 8 for experiments with different weighting on the set of geostrophic equations. A comparison is also made with the root mean square advective flow of water. We note that the turbulent exchange rates on the average are significantly smaller than the advective flows, being merely between 25% and 60% of these. It is interesting that the least value is obtained when the geostrophic condition is relaxed. Without that condition, obviously more freedom is given to find solutions in which the advective flow can better satisfy the condition of itiinimum error for the tracer continuity equations (cf. further section 16). A closer inspection of the fields of turbulence shows that turbulent transfer primarily is of importance for exchange between boxes at the sea surface and adjacent boxes. When large weights are given to the 119 Table 8. Comparison of the average magnitude of advective flow and turbulent exchange between adjacent boxes in experiments with weights 0.1, 0.5, 1.0, 4.0, 8.0 and 50.0 applied to the geostrophic equations. Geostrophic weight 0.1 0.5 1 4 8 50 Mean advective flow (1012 ta-1) 146 100 92 150 175 212 Mean turbulent exchange (1012 ta-1) 37 44 54 70 71 85 Relative magnitude of the mean turbulent exchange to 0.25 0.44 0.59 0.47 0.41 0.40 the advective flow 120 geostrophic equations and thus less freedom to achieve tracer continuity by advective flow, intense turbulent exchange appears spotwise in the solution involving the boxes (region 2, layer 6) and (region 6, layer 6). We note that in their vicinity the tracer gradients are small and accordingly the magnitude of the turbulent exchange unreliable. We recall here the difference between the advective and turbulent components of the solution in that the former also must satisfy the water continuity equation, -while the latter are not governed by any similar condition. These findings and also the fact that about half of the turbulent exchange components become zero because of the constraint of being nonnegative, raises the question of the validity of the formulation of turbulent transfer in terms of eddy diffusivity. Such an assumption can also be questioned on theoretical grounds. The problem of how to describe turbulent transfer of matter in the sea certainly warrants further analysis. 15.3.3 New primary production and detritus flux. Figure 15 shows the new primary production in the surface boxes as deduced for the reference case. The total production for the whole Atlantic Ocean is 2.5 10 5 gC a of which 2.2 10 gC a-1 is in organic and 0.3 10 gCa is in inorganic form. This corresponds to 29, 25 and 4 gC a~"nn respectively. About 20% of this total production occurs in the Antarctic region (region 12) and is dependent on intense upwelling motion which closes the meridional circulation that is deduced for the Atlantic basin. In reality this upward motion may partly take place in other parts of the world oceans, which would correspond to water inflow (to region 12) in upper layers and outflow in the deep sea. Such a pattern might markedly reduce the rate of new primary production in 121 Figure 15. Total new primary production (Qj.) and its partitioning between organic (QQ) and inorganic (Q^) cxsrpounds (unit: 1012 g C a"*1) as deduced for the reference case. The lower figures in each region give! the total new primary production per unit area (g^) and its paritipning between organic (g_) and inorganic (g*) compounds (unit: g C a"1 m"2). 122 region 12. The total production referred to above may therefore be an overestimate. Different geostrophic weighting changes the magnitude of the total new primary production, but markedly so only for strong and presumably unrealistic geostrophic constraints (cf. Table 7). We will, on the other hand, find in section 16.3 that the value for new primary production (and accordingly detritus flux into the intermediate and deeper parts of the sea) is much dependent on the presence of turbulent exchanges as formulated in the present model. Not permitting turbulent transfer reduces new primary production to less than half of what is obtained in this reference case. This sensitivity depends on the fact that inclusion of turbulent terms permits more intense vertical transfer of tracer material (C, C, A and P) upwards which is balanced by increased detrital flux downward. The significance of the values we deduce therefore obviously depends on how well the turbulent transfers are described by the present formulation. We have shown in the previous section (15.3.2) that the uncertainties are considerable as will also be further discussed in section 16. Figure 1$ also shows the spatial distribution of the rate of new primary production. Although the details of this distribution are uncertain we notice the rather large rate in the eastern equatorial region (region 7), where upwelling prevails as compared to the western region. This feature has been noticed in almost all experiments which have been performed. The primary production rates are also systematically larger in the southern parts of the Atlantic Ocean, than in the north, presumably at least partly dependent on the direct meridional circulation cell with downwelling in regions 1-4 that characterizes the reference solution. 123 It. is not meaningful to consider the vertical distribution of decomposition of organic detritus and dissolution of inorganic detritus for each region separately. An attempt has, however, been made to deduce the average vertical distributions for the ocean as a whole. The result is shown in Figure 16. We find that 80% of the organic detritus that leaves the surface layers (of about 100 m thickness) is decomposed above 1000 m depth, while the corresponding figure for inorganic detritus is about 65%. The ratio of organic to inorganic detrital flux, being 5.4 at 100 m, has been reduced to 2.8 at 1000 m and is less than 2.0 below about 3000 m. 15.4 The question of indeterminacy in the reference solution We end the description of our reference solution with an investigation into how well-determined its components are. The satisfactory way to explore that question is to vary the assumptions behind the solution, i.e. derivation of tracer concentrations for boxes, boundary conditions, Redfield ratios, etc., in a way that systematically challenges the chemical, biological and oceanographical considerations on which the adopted procedures were built. A series of experiments of this kind is described in section 16. However, as a background to the sensitivity tests, it is useful to know if the solution, as obtained with given input data, is unique, or if there exist other solutions x that satisfy the minimization requirement (12.3) almost equally well. Uniqueness of the solution to a least-square problem Ax » b is equivalent to absence of solutions to the corresponding homogeneous equation Ax= 0. In our case, the question rather is if the length of Ax can be made much smaller for some vectors x than for others (given that x is normalized, say fi x|j = 1). For the reference case solution the condition number is «Axil1, SW.*. * s bo IIAxll l*i H 124 Detritus i lux g m2 yr" 4000 Figure 16. The average vertical flux of organic and inorganic detrital matter as a function of depth as deduced for the regions 1-11 in the reference case (unit g C a""1 in"2). Each box in the regions considered has been located with regard to its depth and vertical extension assuming that their horizontal area is the same as that of the surface box of the region. The mean depth of the Atlantic Ocean in this way becomes about 4000 m. The decomposition and dissolution rates as obtained from the model solution have been allocated to the layers indicated in the graph and the downward flux determined. 125 A range over little more than two powers of ten is hardly serious, and we conclude that ijideterminacy probably is not a major problem, given the available information. However, it is still interesting to see which combinations of unknowns are least well-resolved. A singular value decomposition of A is useful to this end. The norm|lAxi|is minimized if we set x=*v where v is the singular vector of A corresponding to the smallest singular value. We can interpret the components of v as coefficients defining a certain linear combination of the unknowns, and since llAvll is almost zero we may say that the problem is almost not resolvable with regard to the linear combination v. No singular vector should be expected to be a simple combination of just a few unknowns, but will involve all the 429 unknowns of the system. We may, however, see how the total norm (which is one, for any singular vector) is partitioned between the three types of unknowns: advective, turbulent and biological. We can do so for each of the singular vectors, and such an analysis shows that the biological unknowns contribute significantly more than the others for small singular values, as the right part of Figure 17 shows. Conversely, as we consider singular vectors associated with large singular values, we find that more and more of the norm is due to advective and turbulent unknowns. This is illustrated in the left part of Figure 17. In this sense, the advective part of the solution is better resolved than the turbulent or biological part, probably because advective components appear in all equations. The turbulences do not appear in any dynamical equations, and the biological components are furthermore unaffected by the salinity (and enthalpy) balance. 16 Sensitivity studies All sensitivity experiments described in the following subsections were performed with all rows normalized and no subjective weights except 126 Biological x unknowns + # x x X XX to X X d- x' X _ * . ,+ . Turbulences ** x^* . +&.++% 4- + x x I *x x*< + d-1 XX xXX ^xx* CO d X X d" Advectlons X __r_ 100 150 200 250 300 350 400 450 50 Singular component number SyfTVi • Z"*10^.* system has more equations than unknowns, it will still be underdetermined if there existsW singular^ectoT'v suS that Av = o (cf. text). in the present applicatio^a sSgSar veïor consists of «9 components, out of which 101 describe th^Xc^Tpart of the solution; 184, the turbulent part; and 144, the biological^rt ïhe sum £v? is, by definition, one. Hie diagram shows that tS ScSar' vectors v for which UvJ is small (rightpWtof Sï dSgrS^ïeS to have imch of % their norm in the biological part: aaMT^^taS? the symbol x to the top of the diagram) is larger than 144/429 S <^f^J™ the War «^ ^ to the top of the dSgwm). On S SSfÄ -ttö Sin?Ul^ VeCt0rS,f°r ^^ « AvJ is largTtenft^nave michof their^norm in the advective part: Œhe sum! v? taSn overthe advections (the distance from the bottom of the diagrSto SfsStool +) is larger than 101/429 (the level of the lower solid line) ^^ ' 127 weight 4 on geostrophy and a downweighting by 10 applied to temperature and surface box oxygen equations. Comparisons between solutions are provided mainly by linear regressions between the components of different solutions. Convenience is the prime justification for this, since details in the results otherwise distract from the overall conclusions. Summarizing the differences between 429 parameters in two solutions with three numbers (correlation, slope and y-intercept) inevitably has limitations. Advections dominate the regression calculations because of their tendency to be larger than turbulences. When informative, regressions for the three groups of unknowns (advections, turbulences and detritus fluxes) are presented separately. The intercept is close to zero in all cases. The correlations are used to judge the relative importance of different sensitivites of the model. 16.1 Sensitivity to external forcing 16.1.1 Water flux boundary conditions The first test studies the role of water boundary conditions. Since evaporation, precipitation and exchange with adjacent seas affects all tracers these fluxes are expected to be of importance for deducing our model parameters. By altering all boundary water fluxes with a factor 0.8 a new solution (K 8 in Table 9) is acquired. Correlating this solution with the reference solution gives an r-value of 0.994 with a slope of 0.913. Although the water boundary condition is the most important external forcing the impact of a 20% perturbation on the correlation is weak, while the regression slope indicates that the fluxes are reduced, but the decrease is less than the applied perturbation. It might be tempting to conclude that other factors 128 Table 9. Correlation between solutions discussed in the sensitivity- studies section where basic model assumptions are varied. Solution Slope Coefficient of Note determination between X^and^ K 7 0.996 0.999 14C decay decreased 25% K 8 0.913 0.992 Water exchanges * 0.8 K 9 0.971 0.993 Redfield ratios 1:16:140:-172 K 10 0.999 0.998 Redfield ratios 1:16:103:-172 K 11 0.980 0.994 Top 25 m pC02 samples only K 12 1.00 0.991 pCO2=320 ppm in all surface boxes K 13 0.960 0.953 GEOSECS data only K 19 0.96 0.77 Eddy-diffusivity set to zero 129 control the solution so well that perturbations to the boundary conditions are unable to disturb the solution. The perturbation is, however, evenly applied to all fluxes which partly explains the high correlation coefficient. Experiments with randomly applied perturbations to the water boundary conditions would probably show larger sensitivities. 16.1.2 Radiocarbon decay Another important contributor to the right-hand side of our system of equations is the decay of radiocarbon. This process is important since it is in addition to geostrophy the only forcing of the interior of the system. The central role of carbon and C in biochemical ocean modelling further emphasis the importance of analyzing the radiocarbon system carefully. To test the model's sensitivity to C the rate of decay was decreased by 25%. The solution obtained (K 7 in table 9) yields an r-value of 0.999 when correlated with the reference solution (see Figure j£ ). The slope of the regression line is 0.996 indicating that the interior forcing brought about by the radiocarbon decay term has weak influence on the solutions. Inspection of the contribution of the different terms in the 1 C equations once a solution has been acquired reveals that the radioactive decay terms are small compared to the water transport terms. Because of the large contrasts in C content in the arctic and antarctic waters the decay signal is difficult to retrieve in the Atlantic Ocean since most variations can be explained by mixing between the northern and southern waters. This problem has been discussed by Broecker (1979, 1981) but not explored in an explicit model like here. Difficulties with resolving the small radiocarbon variations are further illuminated by a discussion of the error distribution in the current K07 vs. Ref K13 vs. Ref -1 -0.75-0.50-0.25 0 0.25 0.50 0.75 T—!—!—i—r -0.75 -0.50 -0.25 0.25 0.50 Reference case Referance case Figure 18. ïhe cxsrponents of solution K7 (a) and KL3 (b) versus the reference case. For explanation of the experiments see text and table 9. 131 model. A separate paper by Holmen (1987) extends the discussion to general questions about the utility of C for oceanic studies. 16.1.3 Carbon dioxide exchange It is necessary to quantify well carbon dioxide exchange across the air-sea interface in order to make a successful assessment of oceanic CCu uptake. (Broecker and Peng, 1981). To test the model's sensitivity to air-sea exchange processes three sets of surface box pC02 distributions were used employing different but simple strategies when assigning the pC02 values. By calculating the pCCu in each box from the average DIC, alkalinity, temperature and salinity values assigned to the box the reference case CCu flux boundary condition was acquired. Since our boxes are large, water within one box may both reach the surface and extend to considerable depth. Consequently the pC02 values calculated from box average DIC and alkalinity tend to be high (especially true for high latitude boxes). Therefore a set of pCCu. values averaging only the surface samples (top 25 m) of each box were computed. These values differ significantly from the reference case as shown by the carbon and C fluxes in tables 10 and 11. The surface pC02 values were used to obtain solution K 11 (table 9). The last approach to assigning pC02 values used is to apply a constant pC02 to all surface boxes; this was done to derive solution K 12 where 320 ppm(v) was used exclusively. Solution K 11 gives an r-value of 0.994 when correlated to the reference case and solution K 12 gives 0.991, with regression slopes of 0.98 and 1.00 respectively. These values indicate that the model is insensitive to variations in the C02 flux boundary condition. The changes that do appear are restricted to surface regions. It remains unclear whether the solutions are useful for C02 uptake studies or not. This question is 132 Table 10. Net air-sea carbon and radiocarbon fluxes when pCCu is calculated from surface samples (top 25 m). Positive values are fluxes into ocean. Units are 1013 moles/a for carbon and moles/a for radiocarbon. 00? fluxes Region 8 9 10 11 12 1 -.85 -.57 -1.44 -.94 -.17 1.09 2 .10 -.11 -.01 .91 .34 3 .59 .38 -.12 -.37 4 1.13 .14 -.04 2.16 5 .31 -.08 -.00 -.52 6 .34 -.07 1.76 .64 .34 -.75 -.68 -1.44 -.94 -.18 -1.09 .79 -.03 1.57 C fluxes Region 8 9 10 11 12 1 -2.0 -1.5 -9.7 -.1 4.0 -6.4 2 1.8 1.6 .2 15.3 7.5 3 9.4 8.5 i.o -2.2 4 14.8 2.3 -.3 40.2 5 4.0 -.5 -.0 -3.6 6 4.6 -.4 23.4 11.1 8.2 -.2 .1 -9.7 -.1 4.2 -6.4 16.3 5.2 36.1 133 Table 11. Net air-sea carbon and radiocarbon fluxes when pC02 is calculated from average DIC, alkalinity, salinity and température. C02 fluxes Region 123456789 10 11 12 1 -.21 -.25 -1.10 -1.63 .44 -.12 2 .01 -.17 .02 1.13 .52 3 .43 .03 .01 -.23 4 1.29 -.06 -.03 .34 5 .06 -.05 -.00 -.42 6 .07 -.06 1.42 .32 -.00 -.20 -.42 -1.10 -1.63 .45 -.12 1.13 .29 -.15 C fluxes Region 123456789 10 11 12 1 5.0 1.9 -6.1 -7.8 10.7 4.4 2 .98 1.0 .4 17.6 9.4 3 7.6 4.5 2.3 -.7 4 16.6 .1 -.2 20.9 5 1.3 -.2 -.0 -2.7 6 1.6 -.3 19.5 7.5 4.4 5.9 2.9 -6.1 -7.8 11.2 4.4 20.0 8.7 17.9 134 penetrated in the error analysis section and in further studies by Holmen (1987). Experiments with varied air-sea exchange rates are superfluous in view of the above results. 16.2 Sensitivity to tracer data uncertainties To study the model's response to variations in the tracer data a solution using only GEOSECS data was obtained and compared with the reference solution. The two datasets differ only in the boxes where TTO and GEOSECS data overlap. The radiocarbon values are identical, as well as tracer values in the South Atlantic regions. Consequently only 266 of the 588 (7*84) tracer values differ. The differences between the two datasets appear small when inspecting Figure 19 showing the box values for all tracers. When correlating the components of a solution obtained using only GEOSECS data (K 13 in Table 9) with the reference solution an r-value of 0.95 and slope of 0.96 are acquired. The correlation is lower than the values obtained in the other sensitivity studies showing that the model is more dependent on small scale differences in the tracer fields than on the studied changes of boundary conditions. To illustrate the stronger sensitivity Figure 19 shows the components of solution K 13 plotted versus the base case. The C experiment (K 7) is shown for comparison. Problems related to the représentâtivity of the available datasets for a study of this kind are discussed in section 16.5. Correlation calculations for the three groups of unknowns for experiment K 13 reveal some interesting information. The advections are least affected with an r-value of 0.97 whereas turbulences and detritus have r-values of 0.91 and 0.93, respectively. This is a result of the subjective weight (4) applied to geostrophy. A set of experiments were performed with different weights on geostrophy to obtain solution pairs with the two datasets. Table 12 shows the correlation coefficients for DIC ALK lO CM S • f O / C\2 + ci + ? 10 oi" « • O o FH *i_ + CO N + + 00 + + Q + m0* 2 + + f + + tu rn " OÎ • "T ' 1 I 1 ! r-'-l- "T»— 1.95 2.05 2.10 2.15 2.20 2.25 2.35 D86G0K D86G0K Figure 19. The tracer values for all boxes in the two data-sets D86T0K ara D86GOK. D86GOK utilizes only GE0SECS data whereas D86TOK utilizes TTO/NAS and TTO/TAS data as veil. 136 O eu 9'T T 20X98(1 oT 92*0 OC'0 gs*o 02*0 gro OT'0 90*0 20X99Q 137 03 92 t2 22 02 8T 9Ï fl Z\ 0T 9 201,99(1 »q -CD CO _ •J + <; CO + D86G0 K st 3 5 35. «3 CO + + -^ i ^ 1 rco 9'9S 98 Q'SS Q8 9'tS te 20199(1 138 Table 12. Correlation coefficients for solutions obtained with the complete datas and solutions using GEQSECS data only. Geostrophic Corr. over Corr. of Corr. of Corr. of weight all components adv. terms turb. terms detritus terms 0.1 0.93 0.93 0.95 0.96 0-5 0.93 0.92 0.96 0.99 1«0 0.95 0.96 0.96 0.99 4.0 0.95 0.97 0.91 0.97 8-0 0.95 0.97 0.89 0.96 50.0 0.94 0.98 0.85 0.96 Table 13. Correlation between solutions where the tracer data is rounded off to varied number of decimal places. The reference solution uses data with five decimals. Solution Corr.coeff. Corr.coeff. Corr. coeff. Corr.coeff. (number of for all terms for adv. terms for turb.terms for detr.terms decimal places) Four 1.00 1.00 1.00 1.00 Three 1.00 1.00 1.00 1.00 Two 0.98 0.99 0.95 0.99 One 0.81 0.87 0.61 0.83 139 these experiments. Advections have high correlations when geostrophy is upweighted, whereas the turbulent coefficients have relatively low correlations. This shows, again, the influence of geostrophy on maintaining the flowpatterns obtained in the base case. Another sensitivity study performed is intended to explore whether the formulation is numerically sound. The following experiments illuminate the sensitivity to round-off errors in the tracer data. The tracer data utilized to construct the A matrix are given with five decimals which of course is far more than any of the analytical techniques would warrant. By successively rounding off the tracer data to fewer decimal places a set of solutions are created which subsequently are correlated with the base case. Inspection of Table 13 reveals that the system does not give artifical results from the last digits on the input data by showing undue sensitivity. The fact that the solutions essentially remain unchanged even when only two decimals are retained is however problematic. The third decimal obviously does not contain information that is important for determining the components of the solution. Nevertheless this is the level where some of the important processes modelled (examples described below include air-sea exchange of C02, 14 organic detritus, Ca003 detritus, and C decay) are expected to produce concentration changes for boxes with the current topology. The reasons for this weak sensitivity are probably related to the high weight on geostrophy, the large inconsistencies discussed in the error analysis section, or inadequacies in the data used. 16.3 Sensitivity to model assumptions Two fundamentally different studies involving variations of model assumptions were performed. First a set of experiments varying the 140 "Redfield ratios" is described followed by a group of experiments without turbulent coefficients amongst the unknowns. A central assumption in the model is adopting "Redfield ratios" to tie the equations describing the biologically active tracers together. An essential sensitivity study is therefore to study the effects of assumptions about these ratios. The most widely cited values are those of Redfield et al. (1963) with P:N:C:02 ratios of 1:16:106:-138. These are used in the reference solution. Recently Takahashi et al. (1985) and Broecker et al. (1985a) have produced and tested new estimates based on the GEOSECS and TTO data. These authors give two estimates depending on different assumptions regarding the oxidation state of carbon in the organic detritus namely 1:16:140:-172 and 1:16:103:-172. The alternative ratios where used for solutions K 9 and K 10 respectively (cf. Table 9). Solution K 9 correlates with the reference solution giving an r-value of 0.993 and K 10 gives 0.998 with the slopes 0.97 and 1.00. The model is thus insensitive to the Redfield ratios at the level discussed as significant in the literature. No attempts to find "best" estimates of the Redfield ratios have been undertaken due to the insensitivity. The usefulness and significance of the Redfield ratios appearing in the literature is discussed by Holmen (1987) as a follow-up of the above results. Frequently box-model studies with resemblances to the current model exclude the role of turbulence (see, for example, Wunsch, 1984) based on arguments hypothesizing a subordinate role of mixing on the scales employed. In addition, the use of eddy-diffusion terms on the scales employed here is qeustionable. An interesting investigation is therefore to drop the turbulent coefficients and compare a purely advective solution with the reference case. 141 A group of experiments with varied subjective weights on the geostrqphic equations reveal some drastic differences. Table 14 shows the correlations between advective solutions with different geostrophic weights and their corresponding solutions obtained with turbulence. Clearly the change in model formulation yields rather different pictures of the mean flow. As indicated by the correlation calculations some advective features are retained especially when high geostrophic weights are applied. The detritus fluxes however change radically; when the geostrophic weight is four, the total production decreases by about 60% and the organic to inorganic ratio drops from five to two. One further test of the eddy-diffusivity formulation was performed by replacing the non-negativity constraints with lower bound of -1 for the difffusive terms. Forty percent of the eddy-diffusion terms become negative when geostrophy is given the weight 4. Which version of the model that yields the most realistic results is not evident from the experiments performed hitherto. An immediate conclusion is that the solutions are strongly dependent on whether turbulence is included as a process or not. 16.4 Error analysis of the base case In view of the results in the sensitivity studies the fundamental question of how significant the results are makes its explicit appearance. Although the importance of addressing this query is irrefutable many model formulations refrain from venturing into it. The importance of viewing a model as a hypothesis formulation and the neccessity to proceed to hypothesis testing in order to augment faith in any results cannot be overemphasized (Piatt, 1964). The traditional method of approaching this step is to utilize one set of data to calibrate the model and then verifying the model with an independent 142 Table 14. Correlation between solutions obtained with turbulence and without turbulence for different geostrophic weights. Geostrophic Corr. coeff. Slope Corr. coeff. Corr. coeff. weight for all for all for advective for detrital terms terms terms terms 0.1 0.62 0.63 0.62 0.63 0.5 0.79 1.22 0.78 0.33 1.0 0.79 1.19 0.79 0.30 4.0 0.77 0.96 0.77 0.34 8.0 0.74 0.87 0.74 0.35 50.0 0.81 0.93 0.81 0.42 143 dataset. Transient tracers were intended to serve as the independent data here, especially tritium, bomb radiocarbon and freons. An analysis of the transient tracers is not undertaken here, because its usefulness is questionable given the differences in tine and space scales. In an overdetermined system, like the present one, the calibration steps and verification steps are in a sense combined. Rather than including all tracers at the onset one could have solved a smaller system and subsequently utilized the remaining tracers for testing the models adequacy. The excessive information in the overdetermined problem is not lost but manifested in errors associated with all equations. Inspection of the error field is, therefore, an important step when striving to assess the compatibility of data, the adequacy of processes included in the model, and the importance of geometry and resolution. The distribution of the errors can also serve as a tool to localize inadequacies (Wunsch, 1985) and some progress can be made along this route. A drawback with the least-squares method is, however, its tendency to spread the total error amongst all equations leaving weak traces in the error field of the presumably few erratic equations one wishes to localize and eliminate. A case where all equations are poor is difficult to distinguish from a case where only a few equations are misinterpretations of reality. The complete error field of the normalized system of equations for the base case is shown in Figure 20. The small values for temperature result from the subjective weights applied to this group of equations in the base case. Comparing this figure with Figure 21, which shows the distribution of b the fact that the errors are evenly distributed can 144 Distribution of Ax—b 252 336 420 Equation number vimire 20 The complete error field (A x - b) for the reference case. Srgacer e^tïonf and geostrophic equations are shown wxth thexr normalized values. 145 Distribution of b CO o DIC C14 ALK PHO OXY SAL TEMP + GEO 0.0 2 + • 0 9- + h ^ o + *+* £ + g + v +. + 1° «ta- ä ° + S-/ + * + • + O o- + + «o d~ + + -0.0 3 - 84 168 252 336 420 504 588 Equation number Figure 31,1 îhe distribution of b in normalized units for all equations. 146 readily be seen. Turning to the normalized equations that depict the carbon and 14C equations, Figure 22 shows the distribution of errors for this group alone. No obvious structures are apparent in the figure and despite attempts to sort equations in different categories (boxes with water exchange, boxes with air-sea exchange etc.) no significant signals have appeared. When the equations are converted back to their unnormalized form some interesting structures do appear (Figure 23). Because most large tracer gradients are found in surface regions of the ocean the normalization procedure has a tendency to upweight the deep boxes relative to the surface. This shifts the largest errors into the surface boxes. In order to judge how severe the errors are it is convenient to express them as a change in concentration per year. Since the surface boxes generally have the smallest volumes their errors in carbon turn-over are magnified versus the errors in the deep ocean boxes. Fxgure 24 shows the errors expressed in mol/ (ma) for DIC and DI C. The magnitude of the dissolved inorganic carbon change expected from the whole anthropogenic perturbation during the past 150 years is about 0.05 mol/m . This is a small change compared to the errors in our surface boxes which are around 0.005 mol/m a. The model performs poorly in the region that is most important to depict well when studying the oceanic 002 uptake. This is partly due to the normalization of the system of equations but certainly also due to problems arising in the model formulation itself. Errors in all other tracers are likewise large in surface regions and particularly when compared with expected concentration changes from some of the processes modelled. Table 15 summarizes this error analysis for the remaining tracers. We conclude that there are inconsistencies in the system of equations that obstruct the intended carbon dioxide uptake studies. Some 147 Ax-b for DIC and C14 » * + + + DIC ei4 3. 0 + + + ++ o + + 4- + + q •* • • + J. + J.+ 4?"+ i—i »-*~ * + * ++ + 4 + + + 4- 4+ + + + + tsi ++ ++ »—I <• •*+ + + 4 + + + 4-+ + CO + -1. 0 0. + + + . + + + + 4- + + + o 2. 0 1 + o co- + 4- 1 4- + o 1 o • irf f 1 ' 84 168 Equation number Fiqure 22, Ohe error (Ax-b) for the total carbon and radiocarbon equations in normalized units. Surface boxes are the leftmost equations in each group. 148 Ax-b for DIC and C14 o + DIC C14 + r-i + + O + d~ + \ +*+ • + + -H- + + 0.0 0 l.* J" J. + «- > + + + + + + + + + dO " + + 1 + •H- N O d~ + 1 ¥• + CO o d 1 """1 84 168 Equation number Figure 23. The errors for the total carbon and radiocarbon equations in uraiormalized units. Units are for DIC lO10 moles/a and for 14C 103 moles/a. 149 dC/dt for PIC and C14 o. DIC C14 d w o, U Ö' CO + + + + + _Q »* * J.44.+ Q N V o d" I q d' I «o q d 84 168 Equation number piquye 341 She errors for the total carbon and radiocarbon equations expressed as noles/m3 a for DIC and lcT12 moles/m5 a for"c. 150 Table 15. Analysis of the absolute values of the errors for the equations for each tracer. Group of Mean Standard Minimum Maximum Units equations error deviation error error DIC -0.00051 0.0094 -0.058 0.040 mol/m3a 14 DI C -0.00011 0.0071 -0.025 0.045 10~12mol/m3i Alk -0.00065 0.0038 -0.022 0.008 equ/ma PO4 -0.0042 0.0658 -0.332 0.332 mmol/m a °2 -0000055 0.0201 -0.107 0.052 mol/m a Sal -0.0048 0.110 -0.202 0.739 per mille/a Temp -0.087 1.646 -8.83 4.84 degrees/a 151 origins of these inconsistencies are localized in the following section. Strategies for relieving the localized problems are also suggested. 16.5 The steady-state assumption and representativity of data 16.5.1 Scale considerations When developing models there are a number of principal problems that ittust be considered (Bolin et al., 1981) if the formulation is to be fruitful. The central question is which degree of simplification to choose. The degree of simplification is primarily chosen based on the questions one asks and the types of data available. In numerical approaches, computer limitations can dictate bounds on the complexity of the model. Both calibration and validation demand data with qualities comparable to the complexity of the model; consideration of the feasability of a particular model study based on the available data is necessary. When discussing model complexity and data it is important to note that both spatial and temporal scales must be considered. Converting arguments about space into arguments about time or vice versa is a hazardous task, since they are seldom independent and the conversions often involve implicit assumptions. Frequently a distinction between steady-state and transient data is made, the steady-state tracers are assumed invariant with time for the constructed model, whereas the transient tracers show temporal variations that the model ought to reproduce. These principal problems are all related to the general questions about tracer fields and their modelling. Which spatial scales and temporal scales do the tracer fields contain information about? Are these classes of scales compatible between the different tracers used in the 152 calibration and verification steps? Is the model successful in simulating these scales? To address these questions in a more specific way the following analysis compares the scale problems for the 12-box model (Bolin et al, 1983) and the present 84-box model. Propitious results for simultaneous tracer modelling were obtained with the 12-box model. This lead to the formulation of the 84-box model which should be viewed as a bold attempt to utilize a large portion of the GEOSECS data for diagnostic purposes. Warnings are frequently raised about using geochemical data for high resolution box models (Keeling and Bolin, 1967; Wunsch, 1978) mainly referring to problems related to data representativity problems. Spatial and temporal representativity problems for the two models are penetrated with the aid of the GEOSECS datset and the denser TTO datasets. (Figure 7) The GEOSECS stations are along two north-south lines on either side of the mid-Atlantic ridge. The TTO program contains east-west sections at several latitudes as well. Given the two sources of geochemical data questions related to their utility in modelling efforts of this kind can be posed. Are the gradients used in the finite difference formulation significant? Is there any indication that the assumed steady-state tracers have a temporal variation? Does the denser sampling in TTO show spatial variabilities in tracer fields that where unresolvable in the GEOSECS material, and how are such features to be considered. Variabilities can serve as a guideline to identifying processes that were not included in the model. Identification of inadequacies in our model formulation on the basis of the TTO data set is of ijmportance in that we learn about the mechanisms controlling the distribution of chemical species in the ocean which goes beyond the prevailing broad arguments that are based on simple few parameter models (i.e. simple box T.53 models, end member mixing models). Particularly valuable for developing the ideas about data utility is the work by Enting (1985) and Enting and Pearman (1986). When considering tracer distributions to be used in a model it has proved helpful to systematize their features with regard to those spatial and temporal scales that the model is intended to resolve and treat explicitly (long), and those that have to be considered implicitly (short). Categories of processes as related to the model formulation Spatial scale of the Temporal scale of the process process A IÛNG LDNG B SHORT SHORT C IONG SHORT D SHORT IDNG Using the framework of the 84-box model we can find examples of processes that fall within each category and furthermore discuss the kinds of difficulties that arise. The spatial scales are given by the box topology. One year temporal resolution is sought to permit the intended comparison between model results and for example the Mauna Loa carbon dioxide record (Bacastow and Keeling, 1981). Category h, If there are large scale features of the ocean circulation that change slowly the model assumption of a steady-state is not valid and yields inconsistencies in a diagnostic model. There are indeed observations that show variations on a decadal time scale. (Brewer et al., 1983; Roemmich and Wunsch, 1984). The observed variations in salinity (0.02% ) and temperature (0.2 °C) indicate that this category 154 might be a problem for the current model. Although the observed changes are small they are of the same order of magnitude as many interbox differences. It is interesting to compare the situation for the 84-box model with the 12-box model where the dimensions of the boxes are significantly larger. Although salinity and temperature were not used in the 12-box study we note that the salinity and temperature features on the 12-box model scale are more pronounced than for the 84-box model. Hence the determination of spatial differences in temperature and salinity would not pose a problem when determining inter-box differences for the 12-box model. Garçon and Minster (1987) have introduced temperature and salinity into the 12-box model. Category A processes pose no problems in their analysis. Particular attention should, however, be given to DIC which will be done later in this section. The 14c variations are analyzed in Holmen (1987). We conclude that some inconsistencies in the 84-box formulation may originate from data made unrepresentative by large scale processes. We also conclude that the cruder 12-box model suffers less from the known decadal variations in the tracers used. Processes in category B (short time and length scales) are visualized as eddies and similar phenomena. Given a temporally long enough data set the effects of these processes probably can be averaged out and possibly depicted by, for example, an eddy diffusivity term. However, there are several stumbling blocks, the first obviously being the availability of data to eliminate these effects. In Figure 19 the tracer concentrations in each of the boxes are shown both as derived from GBOSECS only and the combined TTO and GBOSECS data, ïhis variability was shown to be the data uncertainty that effected the 155 solutions most in the sensitivity studies. Again the comparison with 12-box model is of interest; the larger number of samples and the greater range of sampling dates within each box as well as the larger differences between adjacent boxes ensure that this problem does not have the same dignity in that case. Even a long data set does not mean that the process readily lends itself to description with eddy diffusivity terms. Eddies of Mediterranean Sea water can propagate across the whole Atlantic and not dissipate until they reach the rigid western boundaries. Thus these eddies effectively "tunnel" tracer material through an otherwise apparently homogeneous region of the ocean (McDowell and Rossby, 1978); see also Category D. An implicit assumption made when stating that the variations can be averaged out is that the real ocean responds linearly to the variations. The well-known summer bias in the data sets are particularly cumbersome in these regard since the periods of deep water formation, and thus the periods when a tracer signal can influence the deep, are believed to be sporadic events during the winter. (Clarke and Gascard, 1983; Gascard and Clarke, 1983; Bennett et al., 1985) As a matter of fact the inadequacy of the model in this regard led to downweighting the temperature equations. This conclusion was based on considerations on the dynamics of penetrative convection. The oceans response to heating and cooling is too complicated to be adequately described by the set of simple processes employed in the present 84-box model. Category C involves long length scales and short time scales; examples could be tides and diurnal variations of sunlight. This category could again be averaged out of the data provided a long enough data set is available. There are a multitude of processes that certainly fall within this category, nevertheless no problems in the current formulation have 156 been identifies* as related to processes of this kind. The mentioned examples are of a regular nature and effective on such a short time scale that they apparently pose little problems in the type of data used here. It can be argued that seasonal cycles that explain part of the variations seen in Figure 19 belong in this category rather than category B. The data show sampling problems which are restricted to regions with little covariance between regions which guided their placement in B. The final category D comprising short length scales but long time scales is in a sense the most interesting. It includes unresolved features that in turn can be interpreted as missing processes in the model. Any permanent structure in the data for a tracer that is not explicitly modelled must be inspected in order to judge whether they hide a phenomenon that has significant influence on the surrounding model regions. The most obvious feature in this context is the existence of boundary currents, especially the western boundary currents. When studying the graphic representation of the tracers used several features that are unresolvable in the current model are readily seen (Figure 25). The figures are along the section of TTO/NAS stations T 127 - T 140 and T 172 - T 200 stretching from Scotland to Greenland and from Greenland to the coast of labrador. Although this section cuts through the northern parts of regions 2 and 3 the features along box boundaries to region 1 are presumably similar. The outstanding features are the East-Greenland currents, the Labrador current and the deep waters on the eastern side of Greenland and at the bottom of the Labrador Sea, which all prove difficult to include in the current formulation. Figure 25, East-west sections from Scotland to Greenland (TTO/NAS stations 127-140) and Greenland to Scotland (TTO/NAS stations 172-200) shewing the distribution of the tracers utilized and density, a) density (sigma-theta), b) temperature, c) salinity, d) dissolved inorganic carbon, e) alkalinity, f) dissolved oxygen, g) phosphate. CONTOURS OF P04 NORTHERN ATLANTIC OCEAN ABOVE 1.25 1.20 1.25 1.15 1.20 1.10 1.15 1.05 1.10 1.00 1.05 0.95 1.00 0.90 0.95 0.85 0.90 0.80 0.85 0.75 0.80 0.70 0.75 0.65 0.70 0.60 0.65 0.55 0.60 0.50 0.55 0.45 0.50 0.40 0.45 0.35 0.40 0.30 0.35 0.25 0.30 0.25 3500 BELOW -60 -50 -40 -30 -20 LONGITUDE CONTOURS OF DENS NORTHERN ATLANTIC OCEAN ABOVE 28.000 27.950 28,000 27.900 27.950 27.650 27,900 27.B00 27.850 27.750 27.800 27.700 27.750 27.650 27.700 27.600 27.650 27.550 27.600 27.500 27.550 27.450 27.500 27.400 27.450 27.350 27.400 27.300 27.350 27.250 27.300 27.200 27.250 27.150 27.200 27.100 27.150 27.050 27.100 27.000 27.050 BELOW 27.000 3500 -60 50 -40 -30 LONGITUDE CONTOURS OF TEMP NORTHERN ATLANTIC OCEAN ABOVE 10.0 9.5 - 10.0 9.0 - 9.5 8.5 - 9.0 8.0 - 8.5 7.5 - 8.0 7.0 - 7.5 6.5 - 7.0 6.0 - 6.5 5.5 - 6.0 5.0 - 5.5 4.5 - 5.0 4.0 - 4.5 3.5 - 4.0 3.0 - 3.5 2.5 - 3.0 2.0 - 2.5 1.5 - 2.0 1.0 - 1.5 0.5 - 1.0 0.0 - 0.5 3500 BELOW 0.0 -60 -40 -30 LONGITUDE CONTOURS OF SAL NORTHERN ATLANTIC OCEAN ABOVE 35.200 35.150 35.200 35.100 35.150 35.050 35.100 35.000 35.050 34.950 35.000 34.900 34.950 34.850 34.900 HI 34.800 34.850 34.750 34.800 34.700 34.750 34.650 34.700 34.600 34.650 34.550 34.600 34.500 34.550 34.450 34.500 34.400 34.450 34.350 34.400 34.300 34.350 34.250 34.300 34.200 34.250 BELOW 34.200 3500 -60 -50 -40 -30 LONGITUDE CONTOURS OF DIC NORTHERN ATLANTIC OCEAN ABOVE 2.150 2.145 2.150 2.140 2.145 2.135 2.140 2.130 2.135 2.125 2.130 2.120 2.125 2.115 2.120 2.110 2.115 2.105 2.110 2.100 2.105 2.095 2.100 2.090 2.095 2.085 2.090 2.080 2.0B5 2.075 2.080 2.070 2.075 2.065 2.070 2.060 2.065 2.055 2.060 2.050 2.055 2.050 3500 BELOW -60 50 -40 -30 -20 LONGITUDE CONTOURS OF ALK NORTHERN ATLANTIC OCEAN ABOVE 2.310 2.307 2.310 2.304 2.307 2.301 2.304 2.29B 2.301 2.295 2.398 2.292 2.295 2.289 2.292 2.286 2.289 2.283 2.286 2.280 2.263 2.277 2.280 2.274 2.277 2.271 2.274 2.268 2.271 2.265 2.268 2.262 2.265 2.259 2.262 2.256 2.259 2.253 2.256 2.250 2.253 BELOW 2.250 3500 -60 -50 -40 -30 -20 -10 LONGITUDE CONTOURS OF 02 NORTHERN ATLANTIC OCEAN ABOVE 0.320 0.315 0.320 0.310 0.315 0.305 0.310 0.300 0.305 0.295 0.300 0.290 0.295 0.285 0.290 0.280 0.285 0.275 0.280 0.270 0.275 0.285 0.270 0.260 0.265 0.255 0.260 0.250 0.255 0.245 0.250 0.240 0.245 0.235 0.240 0.230 0.235 0.225 0.230 0.220 0.225 BELOW 0.220 3500 -60 -50 -40 -30 -10 LONGITUDE 157 The role of western boundary currents for tracer distributions has become increasingly apparent during the course of this work. Particularly the role of the western boundary undercurrent for deep-water tracer distributions is pronounced. Many studies of nutrient tracers have shown this to be the case (Reid et al., 1977; Minster, 1985). Theoretical studies also demonstrate the importance of western boundary currents for tracer distributions (Musgrave, 1985). Furthermore transient tracers show the rapid propagation of material with these currents. (Jenkins and Rhines, 1980; Weiss et al., 1985) The present 84-box topology does not permit an adequate description of these current systems. It is interesting to note that western boundary currents do not create difficulties in the 12-box model since they never reach through the low latitude boxes. In the 84-box topology on the other hand these currents reach through one box and effect a box beyond without necessarily altering the average concentration in the by-passed box and thus essentially functioning as "tracer tunnels". The omission of boundary currents is a weakness in the 84-box topology but their inclusion was impossible when only the GEOSECS data were available. The TTD data is of basic importance for furthering over knowledge about oceanic tracer dynamics on these scales, which also may well be necessary for carbon cycle studies. The existence of category D processes can be viewed as a violation of the basic box model assumption of well mixed boxes. Keeling (1973) has pointed out that it is not necessary to have well-mixed boxes, provided that the fluxes out of the reservoir is directly proportional to the content of the box. Although the statement is sound, inspection of data fields always lead to hesitance over the choice of box boundaries whenever inhomogenities are found. The main cause for this hesitance is 158 that unless the variability can be understood as sampling artifacts, the inhomogeneity always indicates the importance of some process that needs to be considered. Examples of models that do not follow the "misconception" of well-mixed boxes (Sarmiento, 1985) are plentiful (i.e. Stuiver et al., 1983) but restricted to steady-state interpretations. The total amount of a tracer within the box seldom enters the calculations (except for radioactive tracers). The term box then actually is deceiving since the analysis essentially is a rudimentary grid model where the evaluations are performed at the box wall boundaries. If time dependent studies are undertaken at least the perturbations must be considered "well-mixed" since changes propagate to all walls proportionally and instantaneously. 16.5.2 Do available data resolve the processes considered? The categorization above was done as an aid to identifying inadequacies in the model formulation in particular missing processes and unrepresentativity of data. The opposite situation must also be considered: Are there processes included in the model that are unresolved by the data? This amounts to inspecting for regions with homogeneous tracer distributions which are sub-divided into several boxes. It is always desirable to place box boundaries through regions with strong gradients (regions of maximum information content in the tracers). Compromises must, however, be made when using several tracers simultaneously. If the tracers are "orthogonal" enough (Wunsch, 1978) the combination should nevertheless determine all fluxes well. Tracers are, however, often covariant. Although all tracers used are governed by different sources and sinks the independent information content often proves meager giving rise to indeterminacies in the model results. A clear example of 159 inadequate independent information is the case of calcium carbonate detritus and alkalinity- Takahashi et al. (1980) point out the tight relation between alkalinity and salinity in surface waters obtained from the GEOSECS results. Brewer et al. (1986) show the same results for TID/NAS data. The salinity and alkalinity distributions of all surface samples from GEOSECS, TTO/NAS and TIÖ/TAS appear identical (Figure 26). From the 84-box model perspective this signifies that the alkalinity field contains little information to extract calcium carbonate detritus fluxes from. This is displeasing since the process that presumably makes the alkalinity field contain independent information from salinity is the calcium carbonate detritus flux. To separate the independent information residing in the alkalinity field from the salinity a normalization to a salinity of 35 % is performed. The normalized alkalinity contains some structures, (Figure 26) notably high values in the high-latitude regions and constant values between 40 S and 40 N. The high values of normalized alkalinity in the Antarctic region probably is brought about by strong upwelling of deep-^waters enriched in alkalinity from dissolving Ca003 particles Takahashi et al. (1980). The arctic region undoubtably also is affected by water of deeper origin. The "excess" alkalinity in arctic surface waters is also due to other processes (Jones et al., 1983). The ice-freezing effects (that also should appear in antarctic waters) are generally dismissed now, the consensus being that the alkalinity originates in river water inflow (Tan et al., 1983; Anderson et al., 1985). Whichever is valid, neither is incorporated in the 84-box model and could give rise to incompatibilities. The constant values in warm waters can be understood by the calculations of Smethie et al. (1985) that show that the expected 160 Alkalinity vs. Latitude in + t+ / JiAâhH. + + +^+ . + levi + + + + + + ***&+ . * + + ed + + + + + + + + OÎ- N- "T ' T" 60'S 40 20 0 20 40 60 80*N Latitude S^SSi £ÄS S-^» asÄ^s of season. Salinity vs. Latitude 80*N Latitude Alkalinity (sal=35) vs. Latitude o + + + + + + in £ + + + + + + ci + + + + ++ + +-H-4+ +"**++ + + f + + + $ +ij.+ o + ci + % ++ IT) + + W + ci + o ci. 1 ] 1 1 1 1 1 J 1 1 1 1 • 60'S 40 20 0 20 40 60 80'N Latitude 162 alkalinity changes from the Ca003 detritus fluxes measured with sediment traps is below the resolution of the analytical technique. Thus alkalinity does not allow resolving detritus fluxes on the regional scale of the 84-box model. The alkalinity data do not appear to allow diagnostic modelling much beyond the resolution of the 12-box model. 16.5.3 The problem of data on dissolved inorganic carbon Apart from the représentativité problems due to inadequate spatial and temporal coverage the precision of the analytical methods must be considered. All tracers have been scrutinized in this regard but DIC deserves special attention, being the most problematic one. These problems originate both in the difficult analytical method and in the fact that DIC does not have a steady-state distribution. Dissolved inorganic carbon is obviously perturbed by the anthropogenic (X>2 emissions to the atmosphere. No corrections for this has been attempted although the literature contains several suggestions that could be used to overcome this problem (Brewer, 1978; Chen and Millero, 1979). These approaches have been criticized by several authors (Shiller, 1981; Broecker et al., 1985b). In view of the fact that the anthropogenic changes in DIC probably seldom exceed 15 mmol/m3 in subsurface waters and the fact that the systematic differences between GEOSECS and TTO are similar in size (Bradshaw et al., 1981, Brewer et al., 1986) it seems inappropriate to apply a sophisticated (but inevitably in reality crude) correction scheme. Undoubtably the surface waters are contaminated with anthropogenic C02 and the changes probably are large scale but the uncertainties caused by other sampling problems and analytical difficulties dominate. The C02 remains uncorrected and is therefore a strong argument against using DIC as a steady-state tracer. Further complications arise from the fact that 20 % of the interbox DIC 163 differences are smaller than the analytical precision of the experimental technique. Despite the high quality of the GEOSECS and TTO carbonate data the conclusion is that the 84-box model demands too much of the DIC data. Interestingly, the demand may be too great for two, seemingly contradictory reasons. The boxes are too small, and one is seeking to resolve processes beyond the resolution of the data, or the boxes are too large so that one is losing information contained in the data. Resolving this conflict is one of our future aims, and we are encouraged by the prospects of using spectral models. 16.6 Conclusions of the sensitivity studies The sensitivity studies and error analysis presented here lead to several conclusions about the current 84-box model. Many of the conclusions are applicable to any model employing tracer continuity equations on similar scales. Statements about how to proceed in future studies can also be made. The solution which we derive, describing ocean circulation, turbulent exchange and detritus formation and dissolution is quite insensitive to several important processes that have been included in the model formulation. This was shown to be the case for the rate of radiocarbon decay, air-sea exchange of C02 and C and the values adopted for the Redfield ratios. These insensitivities are probably due to data inadequacies and the following specific characteristics of the model. The stability of some gross features of the circulation pattern are maintained by the upweighted geostrophic equations. This is true in spite of their incompatibility with the tracer equations, which may be due either to inadequate data coverage for the determination of representative box averages for the tracer concentrations or neglect of relevant processes or both. 164 The increased model resolution implies that a good many inter-box differences become insignificant when the analytical methods and sampling density are taken into account. Specific problems of this kind have been discussed for DIC, C, alkalinity, salinity and temperature. The encountered sensitivity with different datasets (GEOSECS and TTO) shows that the choice of resolution in the model must be considered carefully in relation to the data available. Cases of unrepresentative data both in space and time have been localized in the present formulation. Great care must also be excercized to ensure compatibility between boundary conditions and interior tracer fields, which is often difficult to attain. We have further found it particularly important to incorporate a better description of the complex air-sea interaction processes, particularly regarding exchange of heat and C02. There exist processes which have not been included in the model but which are significant for maintaining the tracer distributions in the ocean using the adopted spatial resolution. The western boundary currents are particularly important in this regard. It has been shown that turbulent transfer may well play an important role for the transfer in the oceans, but that the use of the eddy-diffusivity concept on the scales employed is questionable, since many negative values were acquired when the nonnegativity constraint was relaxed. The obtained solutions are thus strongly dependent on the assumed model structure. This has been clearly shown by the experiments with purely advective solutions thus excluding turbulent transfer altogether. This finding is of prime importance to consider whenever interpreting model 165 results, camparing results from different models or usiner results determined by any model. To extract the rich information that resides in tracer fields such as TTO and to proceed with diagnostic models of finer resolution, methods better tuned to bridge the different scales resolved by the different tracers are necessary. Spectral models seem attractive in this regard. Air-sea exchange processes are not easily resolved by a diagnostic model. An advisable step is to use mixed layer models that describe the complex processes which occur, particularly the annual cycle, and that can make effective use of the forthcoming satellite data on wind-speed, surface temperature and other physical parameters. Such models might provide the otherwise cumbersome boundary conditions for interior ocean models. Diagnostic models are well suited and remain competitive with general circulation models for studies of the interior and deep ocean provided there is a compatibility between model structure and the resolution of the available data. 17. Issues in Methodology 17.1 Introduction As we have seen in section 16, we are confronted with a worrisome level of incompatibility. Specifically, the reference solution implies a considerable mismatch in the tracer balances for particular tracers in particular boxes. This mismatch is, as discussed in section 16.5, the consequence of a complex interplay, at differing time and space scales, between the data that is employed and the structure of the model, 166 including the processes that are sought to be resolved. In part, some difficulties that arise in this context concern the basis of the approach adopted; others are rather due to itethodological gaps. In the course of these investigations, we have stumbled repeatedly into some of these gaps and consequently now know them somewhat better. Thus before ending this section and turning to the conclusions, it is useful to discuss some of the methodological issues that we see and our tentative ideas for their exploration. 17.2. Incompatibilities due to inaccurate data: Total Least Squares Incompatibility of equations is caused by two factors: The data are inaccurate and the equations are approximate. Both factors are significant, but some insight can be gained if we first discuss them separately. We therefore ignore, in this subsection, the approximate character of the equations, and assume that all apparent contradictions between equations are ; • " : due to inadequate input data. (We analyze the other factor in section 17.3.) In this context it becomes meaningful to search for an exact solution, and in the search process simultaneously adjust the elements a-.± and b> in computing a solution vector x. Let us reconsider first the least squares problem for the Jj-^v system (A,b) where we remove all convex inequality constraints i.e. G=0 or equivalently (Dg A Dx , Dg b) (17.1) In the interest of simpler notation we set D =1, as we did i" section 12 and work in the X space. We leave the unweighting matrix Dg "visible" since this may also represent the incorporation of subjective weights on 167 the rows (by ntultiplication of the unweighted matrix by an additional positive diagonal matrix) that can follow the unweighting procedure. Such subjective weighting may be of interest in the present context of noise in the data. The unconstrained problem for the system (DE*A,DEb) is simply the problem of finding the vector x (unique since we still assume that A is of full column rank) such that II Dg (Ax-b) || is minimum. An equivalent statement of the problem is to find the vector r in B such that II D£r K is minimal and b+r is in the range of A (the subspace of B which is composed of all vectors of the form Ax, where x is from X ). The solution is then the vector x for which Ax=b+r. As we have noted earlier, if the system is indeterminant, one must impose additional side conditions such as the x of minimum norm or some other characteristic to achieve a unique x (see Wunsch 1984). Using vector algebra, we further note that r=R>-b where P is the orthogonal projection of B onto the range of A (see Bolin et al., 1983 Appendix I). This latter statement of a least squares problem is the one in which it is easiest to formulate the total least squares concept. In this rephrasing, we see that a least squares problem for an incompatible system is simply the determination of the minimal perturbation of the boundary conditions (i.e. ; the vector b) such that the perturbed system is no longer incompatible. This, in effect, is to assume that A is well known and that all "incompatibilities" originate from inadequate boundary conditions. The total least squares method allows consideration of uncertainty in A as well as b. We perturb not only b but also the Jv column vectors in A. In order to define a minimal perturbation we thus need a (CT.+1) -dimensional weighting matrix: 168 W = diag (w1., ..., wjx+1) The total least squares problem for (DgA, Dgb) with distribution weight W (but still without convex constraints) is to miniirdze |J Dg [E : r] wjj F (17.2) E,r subject to bfr being in the range of A4E, where E is any Jb x Jx matrix and r is any J^-dimensional vector in B. Here || • |( F denotes the Frobenius norm (i.e. the square root of the sum of the squares of the matrix elements) and (E:rJ represents the Jbx(Jx+i) augmented matrix. Having determined a irdnimizing pair (E,r), the x that satisfies (A+E)x=bfr is said to solve the least squares problem (Dg A, Dgb) with distribution weight W It is important that W is on the right of p!:rjin 17.2 since matrix multiplication from the right affects the columns on[E:rJand hence E and r can be weighted separately. By adjusting the weights in W, one can shift the balance between adjustments of A by E and changes of b by r. For instance, if one wants to minimize the perturbations of b and allow greater perturbations of the data matrix A, one would increase the penalty (measured by W) when perturbing b (by r) by increasing the weight wjx+1 while lowering the weights w,, .. .wJx. Total least squares algorithms are available (Golub and van Loan (1980) ). However, the methods are hardly adequate for our problem. Our matrix A is sparse, and allowing even small perturbations E of the data matrix A that do not share the same pattern of zeros and nonzeros is fatal for the possibility to interpret the solution physically. Demmel (1986) has treated the situation where an arbitrary submatrix of A is subject to perturbations. Although this is a step in the right 169 direction, it is still not general enough for our application. We also observe that total least squares algorithms do not allow for convex constraints on the solution vector x. We conclude, thus, that our attempts to satisfy the equation system exactly have failed. This is not fatal but it is regrettable, since it makes the study of transient processes problematic. In order to explore the response to an external source for any of the tracers in the model (total carbon and radiocarbon are the obvious candidates) one needs to start from a steady state where those tracers are free from drift caused by residues in the least-squares approximation. 17.3. Incompatibilities due to approximate equations We turn next to the opposite extreme, i.e. we assume that all input data are known with good confidence. Instead, we explain the observed incompatibilities between the equations as entirely due to their approximate character, for example the neglect of minor sources and sinks in the tracer balance expressions. A good example of this situation is the thermal^wind equations. The data base (Levitus, 1982) is comparatively reliable, but on the other hand the motion is partly ageostrophic and we cannot expect equations like (3.7) to be exactly satisfied. A principal lesson to be learned is that it may sometimes be rewarding to neglect some small terms from the equations quite intentionally, since it provides information about how large residues to expect and thus offers a means of deciding when enough weight has been laid on an equation or a group of equations (cf. section 15.2). This is, however, only a rather approximate measure in this context, not knowing how large the ageostrophic flow is. 170 Concerning the tracer balance equations, there is no obvious method to estimate how important neglected processes are. The only measure we take is downweighting of the oxygen equations for boxes facing the atmosphere, as described in sections 8.3 and 12, and, in some experiments, neglecting the turbulence transfer to explore the relative role of advective and turbulent terms to maintain tracer balance (section 16.3). 17.4 The combined effect of inaccurate data and approximate equations Let us now consider two approaches to recasting A (and to some extent b where b depends upon concentration data), within the noise that is in the data, in a manner that reduces the incompatibilities and yet that recognizes the approximate character of the solutions and hence does not demand an exact solution. 17.4.1 Separable Least Squares; An experimental approach The elements of the matrix A are uncertain because they are functions of concentration data q£ the correct values of which are unknown. Using the symbol A(q) to denote that A is a function of q? our problem is to ininimize an expression of the form minimize II A(q)-x - bII (17.3) q, x This is a so-called separable least squared problem. Algorithms for its solution have been developed by Golub and Pereyra (1973), and Kaufmann (1975), see also Björck (1981). Unfortunately, the technique becomes of little use in our application, unless constraints can be set for the numbers q^. There are as many concentration data q- as there are tracer equations (both numbers are equal to the number of boxes times the number of tracers, or 84*7 = 588). The matrix A has 588 rows for tracer 171 balance equations and another 80 rows for the geostrqphy equations. The combined set of unknowns, q and x, has 588 components q^ and 429 independent components x. Since 588 + 429 is greater than 588 + 80, the system becomes grossly underdetermined. It has many exact solutions; however, these solutions may include negative concentrations as well as conflicts with the constraints (G ,h) on x . Thus for the approach to be interesting, constraints must also be imposed both on the concentrations q and on the solution x. However, the need for bounds on q further hampers the employment of algorithms devised to resolve the separable least squares problem in addition to the two severe problems with the Total Least Squares technique (inability to treat constraints on x and inability to conserve the element structure of A as A (q). 17.4.2 Iterative minimization We consider the possibility of iteratively minimizing over x and over q in equation (17.3). The approach is still heuristic but might be valuable. For simplicity in notation we sketch this algorithm for a general constrained system j (A, b), (G, h) 1 based on I boxes and N chemical tracers (in our application Parts III and IV, 1=84 and N=7) and do not involve the complex notation of Part I. (In application, we work with the unweighted loop-based system.) Step 1. Establish highest and lowest values acceptable for each tracer in each box and express these constraints by developing N matrices G71' each of dimension 21x1, and 2I-dimensional vectors h, n=l....N. In the following scheme Step 1 is not repeated, i.e. it is not part of the subsequent iteration process. Step 2. Solve the constrained least squared problem for the system 172 {(A, b), (G, h)}. Step 3. Establish the Ixl transfer matrices k" for each of the N tracers. By a transfer matrix K° we mean the square matrix Kn that arises if the I continuity equations for tracer n , (equation 11.1) are algebraically reformulated to express q instead of x as a vector: where k" is independent of q. Step 4. Solve the N constrained least squares problem for the (square) systems j(Kn, k"), (G11, hn)|n=l,...N where k11 indicates the boundary condition for the n:th tracer and (G11, hn) represents the interval constraints on the data (as given in Step 1). Step 5. With the data set found in Step 4, define a new matrix A (q) and to the extent that b depends upon the tracer concentrations, define a new boundary matrix b. We now repeat the sequence beginning at Step 2 again. In practice, we find that the iteration needs only to be repeated two or three times, depending in part on the size of the "error" intervals in the data set (Step 1). Note that this approach is not unique. If one started step 2 with a matrix other than A but still one formed from data consistent with the error intervals, it is not certain that the resulting two iterative paths would lead to the same solutions. We do not consider this a major difficulty since we begin with the mean values for each box which seems the most reasonable, but the concern does increase if the data intervals are large. Further exploration of the current 84 box model using this iterative approach to reducing the incompatibilities will be pursued in a subsequent publication. 173 17.4.3 Singular value decomposition: A classical approach Because of noise in the data, the information in the matrix A is only partly significant. A standard way to identify and perhaps remove the less certain information is to decompose A in the form A = U S VT (17.4) where U and V are orthogonal matrices of sizes (m x m) and (n x n), and S is of the form S = (17.5) where D is an n x n diagonal matrix. "The least-square solution to the unconstrained problem can then be written x = V S+ UT b (17.6) or, in component form, x.-- V '•V*> .y (i7.7) 1 4- s- 'i J J J Obviously, if s- is a "small" singular value and if there is a substantial component of t> in the "direction" utp, then the j :th term would form a large component in the solution, whereas if the singular value were zero, the j:th term would contribute nothing to the solution. The closeness between these two extremes is the essence of the difficulties associated with small singular values. When there are no convex constraints, it is in principle straight-forward to resolve the difficulties. Simply, establish a logical definition of "small", and 174 eliminate in the sum (17.7) all the singular values that are less than this smallest acceptable value. In determining a potential "cut-off" point for the singular values, one should consider noise in the data. One approach is to establish a new matrix A' by slightly altering the data set, wherein, each element is perturbed within preset confidence intervals about each box average. Having determined this new matrix A , which is, in a sense, not different from the matrix A, in so far as the data is concerned, we compute the norm of A~A* (i.e. its largest singular value) and this is used to establish the lower bound for the singular values. Of course, for the purpose of sensitivity studies one should test higher and lower values for this bound as well as a variety of perturbations. In pursuing the approach, however, we met a difficulty in simtaneously satisfying the convex constraints (G, h). The complication occurs because the calculation requires the full application of the singular value decompositiion (Wunsch, 1978, Wunsch and Minster, 1982, Fiadeiro and Veronis, 1982, Bolin et al., 1983) prior to the onset of the calculation of the solution. Specifically, one begins with the creation of a nonfull rank matrix A^. which is formed through the singular value decomposition of A by a truncation of the diagonal matrix S. Specifically, set Atr=U S^V where S^. is the diagonal matrix of singular values except that below the cut-off point all singular values have been set to zero and hence the associated V-singular vectors form a basis for the null (kernel) of A^_. One can now treat this truncated system by using the techniques that we used in Bolin et al. (1983) for meeting constraints in an indeterminate setting (see also Wunsch and Minster 1982). We use the vectors from the null space to modify the pseudo-inverse solution so that the resulting solution meets the constraints. The appealing character of this approach 175 is that we use only the directions associated with the small singular vectors to form the final solution, but we do not weight highly these directions as one implicitly does when using also the inverse of the small singular values. The disadvantage is that the method does not necessarily lead to a solution. In other words, there may be no linear combination x of vectors from the null space of A^ such that x+A^Js meets the constraints imposed by (G, h ). This circumstance can be avoided by rephrasing the least squares concept in a less restrictive manner. In order to realize this rephrasing we state again the constrained least squares problem in this context. Given (A^b) and (G, h) find the vector x such that i) Gsoh; ii)|j Aj^-b|i^AtlAtrb-bl| and iii) l| x-Aj-jJa I] is minimal (this condition is equivalent to demanding that |) xfj is minimal). In this context one relaxes conditions (ii) and (iii). We do this by introducing the augmented full rank system b "*lr - x ^ 0 where alpha is a positive number. Since this system is full rank we can use our standard procedures (Part I) with (G,h) as before. In the case where the previously discussed nonfull rank (indeterminate) system has a constrained solution, the augmented technique will converge to the same solution if we consider the limit as alpha goes to zero. The disadvantages shared by each of these approaches to small singular values is that they require a reconstruction of A, or a creation of A^, before beginning any analysis. Further noise, through round-off errors, enters the process during this reconstruction and before beginning the analysis. This is particularly dangerous since the reconstruction 176 demands the full singular value decomposition using all three matrices. With large systems (A,b), this full application of the singular value decomposition is to be treated with caution. Thus far using advanced algorithms and 64 bit precision, we have been dissatisfied with the results. Further advances in the SVD algorithms, perhaps, hold more promising results. 18 Conclusions The aim of the present study has been to find out how well we can determine the interplay of physical, chemical and biological processes in the sea from the simultaneous observations of the hydrography and quasi-steady distributions of oceanic tracers. The study shows very clearly that this is a difficult problem. The 12-box model developed previously was successful in yielding a quantitative description of the very largest features of global ocean circulation and biogeochemistry that seemed reasonable. The attempt to develop a model with much better resolution, i.e. an 84-box model for the Atlantic ocean has, however, proved to be difficult. The difficulties are of three kinds: 1) We need to adopt a model structure and a resolution that is adequate to permit a reasonable description of the physical, chemical and biological processes that we wish to describe, but still is managable from a computational point of view. The choice of an 84-box model of course was a compromise between these two requirements. It does not permit the description of important oceanic features, e.g. of boundary currents. Still we end up with a total of 668 simultaneous equations to be solved. The choice of horizontal and vertical partitioning of the Atlantic basin was determined by careful consideration of major features of the tracer fields and circulation patterns as we know them from 177 earlier studies of different kinds. Certainly another choice could have been made, possibly with significantly different results. 2) We need to decide which processes to consider in the model and the assumptions to arrive at a reasonably simple description. The internal processes of advective flow, turbulent flow, the formation of organic and inorganic detritus formation and decomposition are obvious first choices as is the assumption of fixed Redfield ratios and that these are applicable both to formation and destruction of detritus. Similarly the quasi-geostrophic condition imposed everywhere except in the vicinity of the equator seems a natural first dynamical constraint in the solution. More questionable is obviously the formulation of turbulent process by using the eddy diffusivity concept, particularly in view of the well-known role of ocean boundary conditions. The formulation of appropriate boundary processes, particularly air-sea exchange is principally necessary, but most difficult. The simple principles adopted are approximate particularly because of the coarse resolution adopted. 3) Available data need to be analyzed with due regard to both model structure and processes included. Hydrographie data are comparatively plentiful and the geostrophic condition was considered well determined. To explore the role of the quasi-geostrophic balance different weights have been applied. Tracer data from the GBOSECS expeditions do barely have sufficient coverage for our purposes and the TTO data gathered in the early 1980's have only improved the situation in the northern Atlantic. Also the data required to formulate the boundary conditions are in some regards inadequate for our purposes. We have striven for the definition of an overdetermined system to be able to derive solutions by minimizing the error when attempting to satisfy the physical-chemical-biolcgical processes that we have chosen 178 as being most relevant. Only in this way are we able to aoccount for and analyze inadequacies with regard to model configuration, inclusion and description of relevant processes and inadequacies of the data used. As should be clear from this brief qualitative description of the problems encountered in the course of the development of the model, we must not expect final and firm qualitative results in a first attempt of this kind (it should be recognized that Riley (1951) made a similar analysis of the Northern Atlantic, but did not have adequate computational resources to permit an optimizational procedure (using inverse methods) as adopted here). Although these difficulties have been troublesome, our attempts to resolve some of them have been rewarding and led to improved understanding about the sensitivity and reliability of the results and indications about further developments of models of this kind that might be profitable. We may summarize our principal findings more specifically in the following way: 1) Ihe internal distribution of temperature and the prescribed flux of enthalpy between the exterior and the model domain is found to be incompatible with the interior fields of other selected tracers (dissolved inorganic carbon, radiocarbon, alkalinity, phosphorus, oxygen and salinity) and appropriate boundary conditions as well as with the condition of geostrophic flow. It seems plausible that this incompatibility primarily is due to inadequate treatment of the air-sea boundary condition because of spatial and seasonal variation of the heat-exchange with the atmosphere. 2) Even when considereing the system in which enthalpy is no longer used, there is considerable incompatibility between the demands for tracer continuity on one hand and geostrophy on the other. An upweighting of the geostrophic equations by a factor of four yields 179 an ageostrophic component of the advective flow which is about 16%, which this solution is chosen as a base case. In the course of this experimentation sensitivity of the solution to the adopted weighting has become very obvious. The choice of weighting is to a certain degree subjective and can be effectively used for studies of the relevance of different processes and sensitivity of data inadequacies. A final adoption of a weighting matrix naturally also is dependant in the objectives of the study. This conclusion is valid not only for the present analysis but for any use of inverse methods for diagnostic studies. This fact is seldom recognized in analyses of marine biogeochemical models by using inverse methods. The base case is characterised by - Advective flows that resemble the large scale flow as traditionally deduced from hydrographie data - Turbulent exchange that is of prime significance for maintenance of tracer balance in the surface layers but elsewhere generally of less importance than is the advective flow. - Detritus flux from the surface boxes that amount totally to 2.2 1015g a-1 of organic carbon and 0.3 1015g a"1 of organic carbon. About 80% of the decomposition of organic tissue takes place above 1 km depth and 50% dissolution of inorganic detrital matter above that same depth. Although the solution we derive seems plausible, the uncertainties are considerable. The magnitude of the advective flow is not well determined, even if the overall flow pattern is reasonably robust. The turbulence in surface layers is decisive for the characteristic 180 pattern of our solution, which is clearly seen from a comparison with a solution derived by neglecting turbulent exchange altogether. The spatial distribution of turbulence is, however, not reliable, nor the spatial distribution of detrital flux from the surface layers into intermediate and deep waters. In the light of the fact that a large number (about 40%) of the turbulent exchange became negative if the condition of nonnegative values for the coefficient of eddy-diffusivity is relaxed, the formulation of turbulence by the use of the concept of eddy-diffusivity may be questioned. 6) The solutions derived do not satisfy the tracer continuity equations well enough to permit extended computations of the transient behaviour of the system subject to an external disturbance, such as an enhanced inflow of carbon or radiocarbon from the atmosphere, due to man-made emissions. We have simply not yet been able to resolve the question about the role of the Atlantic Ocean for the uptake of excess atmospheric carbon dioxide and radiocarbon. It should, however, also be remarked that results deduced with the aid of simpler ocean models may not either be conclusive since the results depend upon major simplifications, the validity of which has often not been quantitatively verified. 7) The results of the experimentation with the 84-box model developed in this paper has led to a more careful analysis of three basic features of importance. a) air-sea exchange, particularly of 002 and radiocarbon b) adoption of the commonly used Redfield ratios c) the ventilation rate of the intermediate and deep waters of the Atlantic Ocean. 181 The details of this analysis is given in a separate paper by Holmen (1987). With regard to more specific results of a methodological numerical nature we note: 8) A method has been developed whereby the exact fulfilment of water continuity can be accoirplished by the transformation of the solution component that described advective flow into a set of loops. 9) The formulation of our problem yields an cverdetermined set of equations A » x = b , in which both A and b are determined from the tracer distributions, hydrographie data and exchange of tracer material with the environment. The uncertainty of the data makes it desirable to employ the method of total least squares or separable least squares, i.e. permit perturbations in both A and b . Available algoritms did not prove adequate for our problem, because we need to preserve the sparsity of A and to confine x to a convex set. In an attempt to explore the sensitivity of our system to uncertainties in A and b as present, we have developed an iterative, approximate method to adjust A and x alternatively. The method does, however, not yield a unique solution, but does yield information regarding the sensitivity of our solution to uncertainties of the data. 10) It is clear that some equations are more approximate than others. The tendency to spread residues evenly between all the equations, which is intrinsic to least-squares methods, therefore probably implies that the solution does not provide the best attainable correspondence with reality. To a degree, this effect can be counterbalanced by downweighting the equations most likely to be ' 182 erroneous. We also have found it desirable to test alternatives to least squares, e.g. minimization of the L^-norm (the sum of the absolute values of the residues) or hybrid methods between 1^- and I^-minimization. 11) The condition number of the matrix A is about 150, which is small enough to eliminate finite-precision problems during the process of computing. However, since box-to-box gradients are sometimes smaller than analytical accuracy, part of the matrix A is still not significant. We have attempted to filter out the noise by setting the smallest singular values of A to zero, but we have been dissatisfied with the performance of solution routines based on singular value decomposition. Also, the choice of truncation limit remains quite arbitrary. The problem is common to all box models with improved resolution. It is likely that the distinction between significant information and noise would stand out more clearly if the unknowns were defined as Fourier coefficients in a spectral representation. 12) A significant source of incompatibility is the neglect of seasonal differences. Summer-to-^winter variations affect the atmospheric exchanges both for enthalpy and carbon dioxide. Seasonal variations in tracer concentrations correlate with variations in motion patterns, and as a result the transport of material down from the mixed layer is probably not adequately described by the present formulation of advective and turbulent processes. The phenomenon is particularly important for temperature, but the biological tracers are also affected. Probably a rewarding task for future work would be to link the intermediate and deep boxes of the present model to a more realistic description of the mixed layer, paying due attention to seasonality. We are in this context greatly encouraged 183 by the recent work by Glover and Brewer (1987) to disentangle spatial and temporal variations of surface carbon dioxide partial pressure and the potential for the use of satellite-derived ocean color data by Esaias et al. (1986). 19. Acknowledgements This work has been supported by the Swedish Natural Science Research Council (NFR) under contracts E-EG 0223-113, E-EG 0223-115, E-EG 0223-117 and E-EG 0223-118. Support was also received from U.S. Department of Energy through subcontract UNH 85-11 with the University of New Hampshire, under subcontract 19X-27419C with Martin Marietta Energy System, Inc., under contract DE-ACO5-840R21400 with the U.S. DOE. In addition to the funds provided by the Department of Energy, the investigation at the University of New Hampshire has been supported by a grant from the National Aeronautics and Space Administration (NASA) : NAGW-848. We also wish to express our appreciation to Mrs. Marianne Djndström for extensive work in typing and organizing the manuscript and to Mrs. Ulla Jonsson for drafting the figures. 184 20 References Aagard, K. and Greisman, P., 1975: Toward new mass and heat budgets for the Arctic Ocean. J. Geophys. Res. 80, 3821-3827. Anderson, L.G., Coote, A.R. and Jones, E.P., 1985: Nutrients and Alkalinity in the Labrador Sea, J. Geophys. Res., 90, 7355-7360. Bacastow, R.B. and Keeling, CD., 1981: Atmospheric Carbon Dioxide Concentration and the Observed Airborne Fraction, in: Carbon Cycle Modelling, edited by Bolin, B., John Wiley & Sons, 103-112. Baumgartner, A. and Reichel, E., 1975: The world water balance. R. Oldenbourg-Verlag, Munchen, Wien. Bennett, T., Broecker, W. and Hansen, J., eds., 1985: North Atlantic Deep Water Formation, NASA Conference Publication 2367. Björck, Å., 1981: Least Squares Methods in Physics and Engineering. Lectures given in the Academic Training Programme of CERN 1980-1981. CERN 81-16, 61 p. CERN, Geneva. Bolin, B., 1986. How much C02 will remain in the atmosphere? In Bolin B., Döös, B., Warrick, R. and Jäger, J. (Eds) The greenhouse effect, climatic change and ecosystems. SCOPE Report 29, pp 93-155. J. Wiley, Chichester, England. Bolin, B. and Stommel, H., 1961. On the abyssal circulation of the world ocean - Part TV. Deep Sea Res. 8, 95-110. Bolin, B., Keeling, CD., Bacastow, R.B., Björkström, A. and Siegenthaler, U., 1981: Carbon Cycle modelling. SCOPE 16, John Wiley & Sons, pp. 1-28. Bolin, B., Björkström, A., Holmen, K. and Moore, B., 1983: The simultaneous use of tracers for ocean circulation studies. Tellus 35B, 206-236. Bradshaw, A.L., Brewer, P.G., Shafer, D.K. and Williams, R.T., 1981: Measurements of total carbon dioxide and alkalinity by potentiometric titration in the GEOSECS program, Earth and Planetary Science Let., 55, 99-115. Brewer, P.G., 1978: Direct Observation of the Oceanic (X>2 Increase, Geophys, Res. Let., 5, 997-1000. Brewer, P.G., Broecker, W.S., Jenkins, W. J., Rhines, P.B., Rooth, CG., Swift, J.H., Takahashi, T. and Williams, R.T., 1983: A climatic Freshening of the Deep Atlantic North of 50°N over the past 20 Years, Science 222, 1237-1239. Brewer, P.G., Bradshaw, A.L. and Williams, R.T., 1986: Measurements of Total Carbon Dioxide and Alkalinity in the North Atlantic Ocean in 1981, in: The Changing Carbon Cycle A Global Analysis, edited by Trabalka, J.R. and Reichle, D.E., Springer-Verlag, 348-370. Broecker, W.S., 1979: A revised estimate for the radiocarbon age of North Atlantic Deep Water, J. Geophys. Res., 84, 3218-3226. 185 Broecker, W.S., 1981: Geochemical Tracers and Ocean Circulation, in: Evolution of Physical oceanography, edited by Warren, B.A. and Wunsch, C., The MIT Press, 434-460. Broecker, W.S., Gerard, R., Ewing, M. and Heezen, B.C., 1960: Natural radiocarbon in the Atlantic Ocean, J. Geophpys. Res., 65, 2903-2931. Broecker, W.S. and Takahashi, T., 1978: The relationship between lysocline depth and in situ carbonate concentration, Deep Sea Res., 25, 65-95. Broecker, W.S. and Peng, T.-H., 1981: A Strategy for the Development of an Improved Model for the Uptake of Fossil Fuel C02 by the Ocean, in: Carbon Cycle Modelling, edited by Bolin, B., John Wiley & Sons, 223-226. Broecker, W.S. and Peng, T.H., 1982: Tracers in the sea. Iamont-Doherty Geological Observatory, Columbia Univ. N.Y. Broecker, W.S., Takahashi, T. and Takahashi, T., 1985a: Sources and Flow Patterns of Deep-Ocean Waters as Deduced From Potential Temperature, Salinity, and Initial Phosphate Concentration, J. Geophys. Res., 90, 6925-6939. Broecker, W.S., Takahashi, T. and Peng, T.-H., 1985b: Reconstruction of Past Atmospheric C02 Contents from the Chemistry of the Contemporary Ocean: An Evaluation, D0E/0R-857, TR020, United States Department of Energy, 79 pp. Bryden, H.L., 1973: New polynomials for thermal expansion, adiabatic temperature gradient and potential temperature of sea water. Deep Sea Res., 20, 401-408. Bunker, A.F., 1980: Trends of variables and energy fluxes over the Atlantic Ocean from 1948 to 1972. Mon. Weather Rev., 108, 720-732. Chen, C.-T. and Millero, F.J., 1979: Gradual increase of oceanic C02, Nature, 277, 205-206. Clarke, R.A. and Gascard, J.-C., 1983: The Formation of Labrador Sea Water. Part I: Large-scale Processes, J. of Fhysical Ocean., 13, 1764-1778. Demmel, J.W., 1986: The smallest perturbation of a submatrix which lowers the rank and constrained total least squares. Report Courant Inst, of Math. Sciences, New York. Enting, I.G., 1985: Principles of constrained inversion in the calibration of carbon cycle models, Tellus, 37B, 7-27. Enting, I.G. and Pearman, G.I., 1986: The use of Observations in Calibrating and Validating Carbon Cycle Models, in: Changing Carbon Cycle A Global Analysis, edited by Trabalka, J.R. and Reichle, D.E., Springer-Verlag, 425-458. Eppley, R.W. and Peterson, B.J., 1979: Particulate organic matter flux and planktonic new production in the deep ocean. Nature, 282, 677-680. Esias, W.E., Feldman, G.C., Mc Clain, C.R. and Elrod, J,A., 1986: Monthly Satellite-Derived Phytoplankton Pgiment Distribution for the Worth, Atlantic Ocean Basin. EOS 67, 827-832. 186 Esbensen, S.K. and Kushmir, Y., 1981: The heat budget of the global ocean. Climatic Research Institute. Oregon State University. Corvallis, Oregon. Fiadeiro, M.E. and Veronis, G., 1977: On weighted-mean schemes for the finite-difference approximation of the advection diffusion equation. Tellus 29, 512-522. Fiadeiro, M.E. and Veronis, G., 1982: On the determination of absolute velocities in the ocean. J. Mar. Res., 40 (Suppl.), 159-182. Fofonoff, N.P., 1980: Computation of potential temperature of seawater for an arbitrary reference pressure. Deep Sea Res., 24, 489-491. Garçon, V. and Minster, J.-F. 1987: Heat Carbon and Water Fluxes in a 12-box model of the World Ocean, Tellus, in press. Gascard, J.-C. and Clarke, R.A., 1983: The Formation of Labrador Sea Water. Part II: Mesoscale and Smaller-Scale Processes, J. of Physical ocean., 13, 1779-1797. Golub, G.H. and Pereyra, V., 1973: The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM J. Numer. Anal. 10, 413-432. Golub, G.H. and van Loan, CF., 1980: An analysis of the Total Least Squares problem. SIÄM J. Numer. Anal. 17, 883-893. Haskell, K.H. and Hanson, R.J., 1981: An algorithm for linear least squares problems with equality and nonnegativity constraints. Math. Programming, 21, 98-118. Holmen, K., 1987: Modelling the Marine Component of the Carbon cycle. Jenkins, W.J. and Rhines, P.B., 1980: Tritium in the deep North Atlantic Ocean, Nature, 286, 877-880. Jones, E.P., Coote, A.R. and Levy, E.M., 1983: Effect of sea ice meltwater on the alkalinity of seawater, J. of Marine Res., 41, 43-52. Kaufman, L., 1975: variable projection methods for solving separable nonlinear least squares problems. BIT 15, 49-57. Keeling, CD. and Bolin, B., 1967: The simultaneous use of chemical tracers in oceanic studies, I. General theory of reservoir models, Tellus, 19, 566-581. Kempe, S., 1982: Long-term records of the C02 pressure fluctuations in fresh water. In. E.T. Degens (ed) Transport of carbon and minerals in major world rivers, Part 1. Mitt. aus dem Geol.-Paläont. Inst, der Univ. Hamburg. Heft 52, 91-332. Levitus, S., 1982: Climatological Atlas of the World Ocean. (Professional Paper 13). National Océanographie and Atmosphere Administration, Washington, D.C. Lynn, R. J. and Reid, J.L., 1968: Characteristics and circulation of deep and abyssal waters, Deep-sea Res., 15, 577-598. 187 McDowell, S. E. and Rossby, H.T., 1978: Mediterranean Water: Än Intense Mesoscale Eddy off the Bahamas, Science, 202, 1085-1087. Millero, F. J., Chen, C.T., Bradshaw, A., and Schleicher, K., 1980: Anew high pressure equation of state for seawater. Deep Sea Res., 27, 255-264. Minster, J.-F., 1985: The Two-Degree Discontinuity as Explained by Boundary Mixing, J. Geqphys. Res., 90, 8953-8960. Musgrave, D.L., 1985: A Numerical Study of the Roles of Subgyre-Scale Mixing and the Western Boundary Current on Homogenization of a Passive Tracer, J. Geophys. Res., 90, 7037-7043. Montgomery, R.B., 1941: Transport of the Florida current off Habana. J. Marine Research, IV, 198-220. Moore, B. and Björkström, A., 1986: Calibrating ocean models by the constrained inverse method. In: Trabalka, J.R., and Reichle, D.S. (Eds.) The Changing Carbon Cycle: A Global Analysis. Springer-Verlag, New York. Oeschger, H., Siegenthaler, U., Schotterer, U. and Gugelmann, A., 1975: A box diffusion model to study the carbon dioxide exchange in nature. Tellus 27, 168-192. Olbers, D.J., Wenzel, M. and Willebrand, J., 1985: The inference of North Atlantic circulation patterns from climatological hydrographie data. Reviews of Geophysics, 23, 313-356. Piatt, J.R., 1964: Strong Inference, Science, 146, 347-353. Redfield, A.C., Ketchum, B.H. and Richards, F.A., 1963: The influence of organisms on the composition of sea water, in: The Sea, vol. 2, edited by Hill, M.N., Interscience, New York, 26-77. Reid, J.L., Nowlin, W.D. and Patzert, W.C., 1977: On the Characteristics and Circulation of the Soutwestern Atlantic Ocean, J. of Physical Ocean., 7, 62-91. Riley, G.A., 1951. Oxygen, phosphate and nitrate in the Atlantic Ocean. Bull, of the Bingham Océanographie Collection, Yale Univ. XIII, Article 1, New Haven, Conn. USA. Roemmich, D. and Wunsch, C., 1984: Apparent changes in the climatic state of the deep North Atlantic ocean, Nature, 307, 447-450. Sarmiento, J.L., 1986: Modeling Oceanic Transport of Dissolved Constituents, in: The Role of Air-Sea Exchange in Geochemical Cycling, edited by Buat-Menard, P., D. Reidel Publishing Company, 65-82. Schlitzer, R., 1984: Bestimmung der Tiefwasserzirkulation des Ostatlantiks mittels einer Boxmodell - Auswertung von 1 C und anderen Tracerdaten. Thesis Ruprecht-Karls Universitet Heidelberg. 95 pp. Shiller, A.M., 1981: Calculating the oceanic OD2 increase: A need for caution, J. Geophys. Res., 86, 11083-11088. 188 Siegenthaler, U., 1983: Uptake of excess C02 by an outcrop-diffusion model of the ocean. J. Geophys. Res. 88, C6, 3599-3608. Smethie, W.M., Takahashi, T., Chipman, D.W. and Ledwell, J.R., 1985: Gas Exchange and C02 Flux in the Tropical Atlantic Ocean Determined from 222 Rn and pO02 Measurements, J. Geophys. Res., 90, 7005-7022. Stommel, H., 1960: The Gulf Stream. A physical and dynamical description. Cambridge Univ. Press. London. Stommel, H. and Schott, F., 1977: The f -spiral and determination of the absolute velocity field from hydrographie station data. Deep Sea Res., 24, 325-329. Stuiver, M., Quay, P.D. and östlund, H. G., 1983: Abyssal Water Carbon-14 Distribution and the Age of the World Oceans, Science, 219, 849-851. Sverdrup, H.U., Johnson, M.H. and Flemming, R.H., 1942: The oceans, their physics, chemistry and general biology. Prentice Hall. Inc., New York. Tan, F.C., Dyrssen, D. and Strain, P.M., 1983: Sea-ice meltwater and excess alkalinity in the East Greenland Current, Oceanologica Acta, 6, 283-288. Takahashi, T., Broecker, W.S., Werner, S.R. and Bainbridge, A.E., 1980: Carbonate Chemistry of the Surface Waters of the World oceans, in: Isotope marine Chemistry, edited by, Goldberg, E., Hozibe, Y. and Saruhashi, K., Uchida Rokakuho, Tokyo, 291-326. Takahashi, T., Broecker, W.S. and Langer, S., 1985: Redfield Ratio Based on Chemical Data from Isopycnal Surfaces. J. Gecphys. Res., 90, 6907-6924. Weiss, R.F., Bullister, J.L., Gammon, R.H. and Warner, M.J., 1985: Atmospheric chloroflouramethanes in the deep equatorial Atlantic, Nature, 314, 608-610. Wunsch, C., 1978: The general calculation of the North Atlantic west of 50^*7 determined from inverse methods. Rev. Geophys. Space Phys., 16, 583-620. Wunsch, C., 1984: An estimate of the upwelling rate in the equatorial Atlantic based on the distribution of bomb radiocarbon and quasigeostrophic dynamics. J. Geophys. Res. 89, 7971-7978. Wunsch, C., 1985: Can a tracer Field be Inverted for Velocity?, J. Physical Ocean., 15, 1521-1531. Wunsch, C. and Minster, J.-F., 1982: Methods for box models and ocean circulation tracers: Mathematical programming and non-linear inverse theory. J. Geophys. Res., 87, 5647-5662. Wust, G., 1935: Schichtung und Zirkulation des Atlantischen Ozeans: Die Stratosphère des Atlantischen Ozeans. Wiss. Ergeb. Deut. Atl. Exped. Meteor. 6, 1, 109-228. Appendix A: Input data for the model equations Tables Al, A3-A5: Internal data, i.e. data on box volumes, areas of vertical interfaces between boxes, tracer concentrations (including temperature) in the boxes. Tables A2-A3, A6-A10: External data; i.e. areas of air-sea interfaces, prescribe exchanges of water to and from adjacent oceans, and the atmosphere; and tracer dal>. (including temperature) for that water. Data on heat exchange with the atmosphere Data on vertical velocity gradients as prescribed by thermal wind balance. Table A1. Volumes of boxes (1Q15 m3) Region 1 2 3 4 5 6 7 8 9 10 11 12 Total 0.93 0.35 0.78 0.73 0.93 0.53 4.25 2 2.38 1.10 0.38 1.08 1.66 1.21 0.64 0.94 9.39 3 0.55 1.47 3.33 2.99 2.39 3.71 3.43 2.52 2.76 2.72 25.87 4 0.16 0.32 1.14 1.35 1.55 2.18 3.55 2.46 2.04 2.42 2.15 1.33 20.65 5 0.14 0.71 1.13 1.87 1.73 1.67 2.96 2.59 2.09 2.66 2.65 1.33 21.53 6 1.71 6.12 5.68 13.69 9.83 7.25 11.94 8.20 7.44 6.45 6.70 22.33 107.34 7 1.70 1.78 2.88 10.97 7.19 5.31 8.92 5.72 4.86 3.56 3.84 4.37 61.10 8 1.70 0.15 0.90 10.13 5.06 4.50 8.37 7.72 5.92 8.40 6.58 10.30 69.73 Total 5.41 9.63 13.20 44.65 29.80 24.46 41.26 32.71 26.61 26.89 25.58 39.66 319.86 Table A2. Areas of air-sea interfaces (10'^ nr) Region 12 3 4 5 6 7 8 9 10 11 12 Total Layer 1 9.80 4.69 6.47 10.07 7.74 6.44 45.21 2 0.81 2.63 0.31 4.90 4.59 13.24 3 3.59 4.47 1.82 1.74 11.62 4 2.83 0.81 0.16 10.07 13.87 5 0.77 0.39 0.01 0.87 2.04 6 0.89 0.12 1.01 Total 4.49 4.79 4.64 10.61 7.32 6.47 10.07 8.05 6.44 6.72 6.33 11.06 86.99 Table A3. Areas of vertical surfaces (&,fi) between adjacent boxes (column 2), geostrophic mass flux (M_) across vertical surface if assuming no horizontal velocity at the sea surface (column 3) and the corresponding mean geostrophic velocity (column 4). Layer Area Mass f lux Geostrophic , 9 velocity (lCW) (Sverdrup) (mm s_1) Boundary between regions 1 and 3 4 32.4 0.081 2.50 5 135.0 -1.32 -9.78 6 87.1 0.699 8.03 Boundary between regions 2 and 3 3 503.8 -2.51 -4.99 4 377.0 -3.05 -8.09 5 834.0 -6.15 -7.37 6 4670 -95.5 -20.4 7 654.0 -35.0 -53.5 Boundary between regions 2 and 4 3 700.0 0.039 0.06 4 276.0 3.81 13.8 5 292.0 5.42 18.6 6 2760 89.0 32.3 7 1100 35.4 32.2 8 91.2 7.87 86.3 Boundary between regions 3 and 5 3 1220 0.139 0.11 4 407 -0.0591 -0.15 5 452 -0.298 -0.66 6 3600 1.49 0.41 7 1420 -4.09 -2.88 8 639 -1.04 -1.63 Boundary between regions 4 and 5 1 280 -2.95 -10.5 2 668 -7.96 -11.9 3 1560 -20.1 -12.9 4 655 -4.81 -7.3 5 845 -6.60 -7.8 6 6270 -79.1 -12. 1250 -3.97 -3.2 Boundary between regions 4 and 6 1 176 -2.01 -11.4 2 83.1 -1.62 -19.5 3 383 -7.67 -20.0 4 354 -7.25 -20.5 5 332 -6.99 -21.1 6 1800 -26.4 -14.7 7 862 -18.0 -20.9 8 1180 -23.2 -19.7 Boundary between regions 5 and 7 1 340 -1.90 -5.59 2 323 -4.12 -12.8 3 1320 -19.6 -14.9 4 1080 -16.9 -15.7 5 1060 -17.1 -16.1 6 5610 -90.8 -16.2 7 2350 -43.4 -18.5 8 2220 -46.9 -21.1 Boundary between regions 6 and 8 1 334 1.03 3.08 2 150 1.78 11.9 3 782 17.8 22.8 4 809 20.6 25.5 5 656 16.9 25.8 6 3180 100 31.5 7 1410 49.4 35.0 8 2910 124 42.6 Boundary between regions 7 and 9 1 223 1.06 4.75 2 268 2.83 10.6 3 1190 18.0 15.1 4 1140 16.1 14.1 5 1010 13.2 13.1 6 4450 69.6 15.6 7 1650 26.9 16.3 8 2830 42.2 14.9 Boundary between regions 8 and 9 1 280 -2.18 -7.79 2 631 -5.90 -9.35 3 1280 -8.19 -6.40 4 955 -1.57 -1.64 5 1060 2.71 2.56 6 4130 36.7 8.89 7 1740 22.5 12.9 Boundary between regions 8 and 10 2 910 4.52 4.97 3 2020 12.8 6.34 4 1630 11.2 6.87 5 1310 9.00 6.87 6 3950 29.5 7.47 7 1360 16.4 12.1 8 2470 25.3 10.2 Boundary between regions 9 and 11 2 1020 0.449 0.44 3 1490 3.44 2.31 4 1230 5.99 4.87 5 1210 9.60 7.93 6 4120 59.4 14.4 7 2350 47.9 20.4 Boundary between regions 10 and 11 2 77.7 -0.676 -8.70 3 543 -10.2 -18.8 4 583 -21.4 -36.7 5 638 -30.0 -47.0 6 1810 -121 -66.9 7 220 -18.2 -82.0 Boundary between regions 10 and 12 4 1210 5.87 4.85 5 1060 10.0 9.43 6 5330 28.8 5.40 7 647 -0.459 -0.71 8 1860 4.32 2.32 Boundary between regions 11 and 12 4 649 0.922 1.42 5 764 3.66 4.79 6 5420 32.2 5.94 7 555 6.57 11.8 8 1150 19.3 16.8 Table A4. Box averages for DIC,*C, alkalinity, phosphorus, oxygen, salinity and temperature as deduced :rom GEQSECS observations. WESTERN SECTION DIC Region Layer 1 2 4 6 8 10 12 2.044 2.007 2.045 2.074 2.128 2.068 2.047 2.089 2.127 2.183 2.138 2.102 2.051 2.139 2.188 2.211 2.185 2.170 2.144 2.131 2.166 2.200 2.207 2.203 2.213 2.211 2.131 2.173 2.171 2.168 2.169 2.209 2.241 2.131 2.120 2.176 2.171 2.167 2.190 2.245 2.131 2.180 2.193 2.203 2.216 2.240 2.239 EASTERN SECTION DIC Region Layer 1 3 5 7 9 11 12 2.116 1.966 2.032 2.073 2.141 2.093 2.046 2.092 2.127 2.189 2.188 2.113 2.051 2.128 2.183 2.217 2.227 2.183 2.144 2.131 2.141 2.185 2.208 2.216 2.224 2.211 2.131 2.140 2.171 2.167 2.183 2.213 2.241 2.131 2.138 2.181 2.179 2.193 2.209 2.245 2.131 2.142 2.189 2.195 2.215 2.237 2.239 WESTERN SECTION Cl 4 Region Layer I 2 4 6 8 10 12 .952 .940 .952 .942 .936 .942 .932 .952 .936 .918 .920 .920 .952 .940 .928 .894 .896 .900 .890 .947 .935 .914 .892 .880 .880 .876 .942 .930 .910 .906 .898 .872 .844 .936 .925 .906 .900 .894 .878 .842 .930 .920 .900 .880 .858 .850 .840 EASTERN SECTION Cl4 Region Layer 1 3 5 7 9 11 12 .936 .936 .944 .932 .930 .936 .950 .940 .926 .914 .916 .934 .952 .934 .922 .902 .894 .906 .890 .947 .926 .914 .894 .876 .876 .876 .942 .920 .906 .904 .890 .870 .844 .936 .906 .894 .890 .882 .860 .842 .930 .880 .882 .884 .878 .850 .840 WESTERN SECTION ALKALINITY Region Layer 1 2 4 6 8 10 12 2.379 2.366 2.411 2.386 2.354 2.342 2.337 2.282 2.351 2.309 2.296 2.281 2.282 2.285 2.322 2.301 2.301 2.290 2.291 2.282 2.294 2.322 2.310 2.314 2.315 2.316 2.282 2.295 2.318 2.317 2.319 2.330 2.345 2.282 2.271 2.323 2.328 2.328 2.328 2.353 2.282 2.299 2.336 2.346 2.351 2.353 2.351 EASTERN SECTION ALKALINITY Region Layer 1 3 5 7 9 11 12 2.353 2.322 2.368 2.392 2.324 2.322 2.295 2.316 2.342 2.301 2.295 2.284 2.282 2.310 2.315 2.295 2.297 2.302 2.291 2.282 2.304 2.324 2.303 2.308 2.324 2.316 2.282 2.291 2.324 2.311 2.319 2.335 2.345 2.282 2.284 2.334 2.328 2.337 2.342 2.353 2.282 2.287 2.342 2.340 2.353 2.358 2.351 WESTERN SECTION PHOSPHATE Region Layer I 2 4 6 8 10 12 .034 .140 .112 .136 .733 .480 .413 .559 .703 1.763 1.551 1.485 .261 .979 1.500 2.221 2.158 2.133 1.974 .938 1.060 1.555 2.072 2.135 2.289 2.339 .938 1.071 1.226 1.390 1.507 1.990 2.241 .938 1.059 1.242 1.292 1.376 1.712 2.239 .938 .985 1.402 1.712 1.935 2.226 2.221 EASTERN SECTION PHOSPHATE Region Layer 1 3 5 7 9 11 12 .867 .100 .215 .103 1.265 .759 .712 .351 .785 1.939 1.925 1.458 .261 .836 1.555 2.336 2.357 2.186 1.974 .938 1.084 1.599 2.144 2.149 2.322 2.339 .938 1.083 1.349 1.464 1.608 1.932 2.241 .938 .978 1.416 1.439 1.536 1.784 2.239 .938 1.408 1.498 1.493 1.636 2.038 2.221 WESTERN SECTION OXYGEN Region Layer 1 2 4 6 8 10 12 .215 .201 .214 .213 .150 .207 .237 .300 .190 .124 .192 .258 .330 .257 .159 .144 .198 .230 .307 .310 .268 .177 .165 .189 .192 .218 .310 .283 .250 .238 .234 .202 .204 .310 .282 .267 .259 .254 .229 .218 .310 .290 .262 .244 .234 .223 .231 EASTERN SECTION OXYGEN Region Layer 1 3 5 7 9 11 12 .232 .202 .223 .222 .113 .191 .248 .270 .179 .090 .114 .242 .330 .260 .140 .129 .138 .206 .307 .310 .233 .160 .149 .169 .185 .218 .310 .266 .225 .230 .220 .206 .204 .310 .283 .250 .245 .236 .224 .218 .310 .240 .247 .247 .235 .220 .231 WESTERN SECTION SALINITY Region Layer 1 2 4 6 8 10 U 36. 345 36.153 36.762 36.496 35.863 35.682 35.350 34.908 35.821 34.935 34.719 34.427 34.452 34.847 35.160 34.551 34.380 34.279 34.076 34.927 34.928 35.039 34.689 34.561 34.480 34.368 34.927 34.923 35.019 34.940 34.890 34.736 34.651 34.927 34.936 34.934 34.922 34.913 34.843 34.686 34.927 34.915 34.877 34.805 34.750 34.689 34.668 EASTERN SECTION SALINITY Region Layer 1 3 5 7 9 11 12 35.959 35.639 35.943 36.664 35.520 35.429 34.977 35.152 35.948 34.988 34.781 34.544 34.452 35.226 35.318 34.596 34.477 34.368 34.076 34.927 35.170 35.225 34.718 34.603 34.515 34.368 34.927 34.987 35.169 34.936 34.861 34.765 34.651 34.927 34.992 34.937 34.913 34.887 34.820 34.686 34.927 34.906 34.894 34.874 34.854 34.748 34.668 WESTERN SECTION TEMPERATURE Region Layer I 2 4 6 8 10 12 24.852 25.823 23.309 18.389 16.504 15.913 14.591 9.131 13.715 9.512 8.530 6.574 5.636 6.912 8.550 5.179 3.880 3.126 .862 .204 5.745 6.619 4.433 3.336 2.589 1.412 .204 3.463 4.219 3.737 3.282 2.530 .891 .204 2.304 2.555 2.473 2.492 2.259 .299 .204 1.745 1.768 1.187 . .697 .140 -.202 EASTERN SECTION TEMPERATURE Region Layer 1 3 5 7 9 11 12 16.606 26.404 23.002 18.571 15.001 15.208 13.182 10.797 13.941 10.002 8.916 7.679 5.636 8.784 9.336 5.363 4.558 3.830 .862 .204 7.563 7.619 4.706 3.703 2.809 1.412 .204 3.887 5.028 3.577 3.155 2.474 .891 .204 2.146 2.516 2.426 2.357 1.954 .299 .204 2.147 2.038 1.867 1.781 .996 -.202 Tfoble A5. As Table 5 but based on a cantoination of GEOSECS and ITO data. WESTERN SECTION DIC Region Layer 1 2 4 6 8 10 12 1 2.034 2.022 2.045 2 2.081 2.133 2.068 2.047 3 2.084 2.133 2.182 2.138 2.102 4 2.049 2.122 2.179 2.202 2.185 2.170 2.144 5 2.131 2.135 2.181 2.198 2.203 2.213 2.211 6 2.131 2.135 2.149 2.154 2.169 2.209 2.241 7 2.131 2.120 2.143 2.152 2.167 2.190 2.245 8 2.131 2.117 2.148 2.173 2.216 2.240 2.239 EASTERN SECTION DIC Region Layer 1 3 5 7 9 11 12 1 2.064 2.019 2.032 2 2.090 2.154 2.093 2.046 3 2.092 2.128 2.199 2.188 2.113 4 2.049 2.128 2.165 2.216 2.227 2.183 2.144 5 2.131 2.141 2.176 2.205 2.216 2.224 2.211 6 2.131 2.140 2.161 2.159 2.183 2.213 2.241 7 2.131 2.138 2.160 2.158 2.193 2.209 2.245 8 2.131 2.142 2.155 2.162 2.215 2.237 2.239 WESTERN SECTION C14 Region Layer 1 2 4 6 8 10 12 .952 .940 .952 .942 .936 .942 .932 .952 .936 .918 .920 .920 .952 .940 .928 .894 .896 .900 .890 .947 .935 .914 .892 .880 .880 .876 .942 .930 .910 .906 .898 .872 .844 .936 .925 .906 .900 .894 .878 .842 ,930 ,920 ,900 .880 .858 .850 .840 EASTERN SECTION Cl4 Region Layer 1 3 5 7 9 11 12 .936 .936 .944 .932 .930 .936 .950 .940 .926 .914 .916 .934 .952 .934 .922 .902 .894 .905 .890 .947 .926 .914 .894 .876 .876 .876 .942 .920 .906 .904 .890 .870 .844 .936 .906 .894 .890 .882 .860 .842 .930 .880 .882 .884 .878 .850 .840 WESTERN SECTION ALKALINITY Region Layer 1 2 4 6 8 10 12 2.378 2.363 2.411 2.379 2.348 2.342 2.337 2.294 2.340 2.298 2.296 2.281 2.282 2.291 2.310 2.289 2.301 2.290 2.291 2.282 2.295 2.308 2.297 2.314 2.315 2.316 2.282 2.287 2.299 2.297 2.319 2.330 2.345 2.282 2.271 2.294 2.304 2.328 2.328 2.353 2.282 2.264 2.294 2.311 2.351 2.353 2.351 EASTERN SECTION ALKALINITY Region Layer 1 3 5 7 9 11 12 2.401 2.347 2.368 2.394 2.331 2.322 2.295 2.316 2.341 2.299 2.295 2.284 2.282 2.310 2.319 2.288 2,297 2.302 2.291 2.282 2.304 2.324 2.294 2.308 2.324; 2.316 2.282 2.291 2.316 2.296 2.319 2.335 2.345 2.282 2.284 2.310 2.305 2.337 2.342 2.353 2.282 2.287 2.303 2.305 2.353 2.358 2.351 WESTERN SECTION PHOSPHATE Region Layer l 2 4 6 8 10 12 .040 .183 .112 .188 .905 .480 .413 .536 .853 1.797 1.551 1.485 .235 .881 1.619 2.225 2.158 2.133 1.974 .921 1.013 1.566 2.082 2.135 2.289 2.339 .921 1.069 1.226 1.400 1.507 1.990 2.241 .921 1.014 1.221 1.320 1.376 1.712 2.239 .921 1.032 1.359 1.660 1.935 2.226 2.221 EASTERN SECTION PHOSPHATE Region Layer 1 3 5 7 9 11 12 .043 .168 .215 .190 1.220 .759 .712 .511 .828 1.859 1.925 1.458 .235 .890 1.347 2.293 2.357 2.186 1.974 .921 1.049 1.408 2.116 2.149 2.322 2.339 .921 1.104 1.279 1.482 1.608 1.932 2.241 .921 1.208 1.396 1.417 1.536 1.784 2.239 .921 1.408 1.465 1.482 1.636 2.038 2.221 WESTERN SECTION OXYGEN Region Layer 1 2 4 6 8 10 12 .215 .199 .214 .208 .148 .207 .237 .285 .177 .122 .192 .258 .340 .275 .148 .141 .198 .230 .307 .311 .277 .178 .162 .189 .192 .218 .311 .283 .249 .236 .234 .202 .204 .311 .281 .265 .256 .254 .229 .218 .311 .280 .259 .243 .234 .223 .231 EASTERN SECTION OXYGEN Region Layer 1 3 5 7 9 11 12 .223 .195 .223 .211 .111 .191 .248 .256 .177 .089 .114 .242 .340 .243 .159 .115 .138 .206 .307 .311 .234 .165 .144 .169 .185 .218 .311 .258 .215 .225 .220 .206 .204 .311 .259 .244 .246 .236 .224 .218 .311 .240 .242 .245 .235 .220 .231 WESTERN SECTION SALINITY Region Layer 1 2 4 6 8 10 12 36.335 36.150 36.762 36.441 35.830 35.682 35.350 34.688 35.747 34.925 34.719 34.427 34.349 34.726 35.074 34.573 34.380 34.279 34.076 34.922 34.876 35.052 34.688 34.561 34.480 34.368 34.922 34.903 35.010 34.951 34.890 34.736 34.651 34.922 34.921 34.931 34.923 34.913 34.843 34.686 34.922 34.918 34.879 34.818 34.750 34.689 34.668 EASTERN SECTION SALINITY Region Layer 1 3 5 7 9 11 12 36.938 35.916 35.943 36.738 35.713 35.429 34.977 35.319 35.823 35.054 34.781 34.544 34.349 35". 385 35.402 34.669 34.477 34.368 34.076 34.922 35.242 35.420 34.753 34.603 34.515 34.368 34.922 35.068 35.301 34.948 34.861 34.765 34.651 34.922 34.954 34.939 34.918 34.887 34.820 34.686 34.922 34.906 34.893 34.878 34.854 34.748 34.668 WESTERN SECTION TEMPERATURE Region Layer l 2 4 6 8 10 12 23.458 25.922 23.309 18.024 16.312 15.913 14.591 8.288 13.258 9.432 8.530 6.574 4.018 6.081 8.073 5.296 3.880 3.126 .862 .133 5.414 6.592 4.502 3.336 2.589 1.412 .133 3.424 4.193 3.797 3.282 2.530 .891 .133 2.208 2.479 2.466 2.492 2.259 .299 .133 1.965 1.790 1.288 .697 .140 -.202 EASTERN SECTION TEMPERATURE Region Layer 1 3 5 7 9 11 12 22.446 24.936 23.002 19.447 15.785 15.208 13.182 11.448 13.397 10.142 8.916 7.679 4.018 9.161 9.683 5.801 4.558 3.830 .862 .133 7.675 8.506 4.975 3.703 2.809 1.412 .133 4.205 5.865 3.753 3.155 2.474 .891 .133 2.605 2.534 2.442 2.357 1.954 .299 .133 2.147 2.021 1.894 1.781 .996 -.202 Footnote to Tables A4 and A5. The experiments in section 15 were based on a data set which is slightly different from the ones given here. The difference is due to a different formula being used to compute density as function of T and S, a matter of some importance for the question which box to count a given sample to, and, therefore, also for the averages q^. Although there are noticeable differences between the experiments based on these different data sets, the discussion and the conclusions are nowhere significantly influenced by this difference. Table A6. Total net fresh water influx due to precipitation, evaporation and rivers (column 2) and river inflow contribution (column 3). Water influx River inflow Region, layer (1012 ton a-1) (1012 ton a-1) 1.4 1.9 Polar Sea rivers: 1.3 1.5 0.3 2.3 1.3 2.4 0.6 3.3 1.3 European rivers: 0.3 4.1 -8.2 4.2 0.6 St. Lawrence: 0.3 5.1 -5.4 5.2 -1.0 6.1 6.0 Amazonas: 6.9; Niger: 0.3 7.1 -3.8 Zaire: 1.3 8.1 -6.3 South Am. rivers: 0.6 9.1 -7.6 10.2 -0.3 10.3 0.6 11.2 -1.0 11.3 1.0 12.4 3.2 12.5 0.3 Sum -16.5 Polar sea and rivers 11.0 Table A7„. Water exchange between the Atlantic Ocean and the Mediterranean Sea. Flew to the Flew to the layer Atlantic Ocean Mediterranean (1012 ton/a) (1012 ton/a) 2 0 55 3 10 0 4 15 0 5 16 o 6 12 0 Table A8. Water flow from region 6 to region 4 through the Caribbean Sea and the Gulf of Mexico. Layer Flow (1012 ton/a) 1 390 2 180 3 230 4 50 5 25 Table A9. Enthalpy and tracer concentrations in water coming in through Davis Strait and from the Mediterranean Sea. Tracer Import through Import from the Davis Strait Mediterranean Sea .Salinity^ %c 33.400 37.825 Temperature., °C 0.000 13.391 | I Carbon 2.100 2.130 ; mmoles/kg f -55 -50 j j 'Alkalinity 2.230 2.352 :meq/kg I Phosphate 0.8000 0.2004 ' /* moles/kg Oxygen 0.3300 0.2020 < mmoles/kg Table A 10. Enthalpy flux between the atmosphere and the sea deduced from data presented by Bunker (1980) and Esbensen and Kushmir (1981) (1012 y)(Negative numbers denote flux to the atmosphere) Region Layer 1 2 3 4 5 6 7 8 9 10 11 12 1 -490 94 162 402 155 0 2 -81 79 98 138 3 -215 -134 17 : 4 -170 -49 -9 • -101 • 5 -42 -23 -44 6 • -45 -10 Footnote to table AlO As the table shows, the sum of the heat exchanges over the 22 surface boxes is not zero, but rather a net flow of 268 10 2 W to the atmosphere, (which corresponds to about 3 W/nr). Ihis is compensated for when computing the appropriate boundary conditions for enthalpy inflow from the Pacific Ocean. Appendix B: The complete base case solution Table Bl. Mass fluxes (m^j) in the western basin as obtained in the base case solution. The numbers inside the boxes denote the east-vest exchange with the corresponding box in the eastern basin. Unit: 1015 m3 a-1. Table B2. Mass fluxes (nijj) in the eastern basin for the base case solution. Unit 1015 m3 a"1. Table B3. Horizontal and vertical turbulent exchanges in the western basin for the base case solution. The numbers refer to the product Kij -^-:j /^ij expressed in 1015 m3/ a. Numbers inside boxes denote east-west exchanges. Table B4. Same as Table B3, but for the eastern basin. Table B5. Dissolution of biogenetic material (positive number) or new primary production (negative number) in the western basin boxes as obtained in the base c<> solution. In each box, the upper figure stands for organic compounds, the lower figure, for calcium carbonate. Unit: 1014 moles C a-1. Table B6. Same as Table B5, but for the eastern basin. c ?2 Î r,çryWEÏ51-T 4 19:19:35 G3/20/87 WESTÇRM SECTION ADVECTÏONS 2 4 6 8 10 12 S*******E s*******E s******** * 0.055 * 0.014 * 0.134 * 0.243v * 0.099 * * ********** ********** * * EAST * >>>» * WEST * ««< * WEST * ********* ********* ********** 0.317 0.003 0.033 *0.117 $** *S******** * 0.049 * 0.0C4 * 0.016 * 0.093 * 0.283 * 0.032 * 0.146 * * ********** ********** ********** * * EAST * <«<< * WEST * <<«< * WEST * >>>» * EAST * ********** ********* ********* ********* 0.044* 0.408 0.071 0.124 0.231 *OOWN DOWN UP DOWN UP S*******E* ********* ********E ********* s******** * 0.161 * 0.433 $ 0.041 * 0.011 * 0.003 * 0.325 * 0.512 * 0.149 * 0.779 * * ********** ********** ********** ********** * * EAST * <<«< * EAST * <<<<< * EAST * <<«< * WEST * >»>> * EAST * i* V 'C rP *C -• ********* _ ********* ********* ********* ********** 0.38"' 0.170 0.010 0.162 0.723 Si.534 DOWN DOWN DOWN DOWN DOWN UP* S******** ********£ ********* ********* ********* *s*******= * 0.015 * 0.113 * 0.07? * 0.119 * 0.016 * 0.105 * 0.261 * 0.286 • 0.119 * 0.504 * 0.010 * * ********** ********** ********** ********** ********** ********** * * >>»> * EAS? * <<<<< * EAST * <<<<< * EAST * ««< * WEST * >>>>> * EAST * >>>>> * * ********* ********* ********* ********* ********* ********* 0.012 0.366 0.042 0.099 0.068 0.329 0.971 OQWN DOWN DOWN DOWN DOWN DOWN U° S ******** s******** ********* ********E ********* ********* s******* = * * 0.015 * 0.349 * 0.079 * C.116 * 0.015 * 0.265 * 0.190 * 0.259 * 0.042 * 0.341 * 0.197 * * * ********** ********** ********** ********** ********** ********** * * * >>»> * EAST * <<<<< * EAST * <<<« * EAST * ««< * WEST * >>>» * EAST * >>»> * * ********* ********* ********* ********* ********* ********* ********* 0.102 0.111 0.114 0.015 0.095 0.167 0.790 DOWN DOWN U° UP DOWN UP LD S ******** ********* ********* ********* ********* ********* s*******5 * * 0.115 * 0.181 * 0.349 * 0.133 * 0.303 * 0.223 * 0.327 * 0.355 * 0.055 * 0.157 & 0.371 & * ********** ********** $£***,;.£££.«. £$*$££.,.*.}.* ********** ********** * * * >>>>> * Ç*ST * >>>>> * WEST * >»>> * EAST * ««< * WEST * >>>>> * WEST * »>>> * ********* ********* ********* ********* ********* ********* ********* 0.000 3.303 0.065 0.385 0.067 0.325 0.357 UP U° DOWN DOWN DOWN UP UD ********* ********* ********* ********* ********* ********* ********* * * 0.000 * 9.620 * 0.156 * 0.341 * 0.010 * 0.029 * 0.067 * 0.065 * 0.176 * 0.123 * 0.047 * * * ********** ********** ********** ********** ********** ********** * * * * WEST * >>>>> * CAST * <«<< * WEST * «<<< * EAST * >>>>> * WEST * <<<<< * * ********* ********* ********* ********* ********* ********* ********* 0.000 ".161 0.110 0.471 0.241 0.021 0.333 OOWN DOWN UP OOWN UP DOWN U? ********* ********* ********* ********* ********* ********* ********* *********** O.cno ********* 0.00" *********** 0.16>>>>1> ********* 0.000 *********** 0.05>>>>1> ********* 0.16EAS0T *********** 0.36>>>>2> ********* 0.000 *********** 0.12>>>>1> ********* 0.000 *********** 0.14>>>>2> **; :GKT 4 19:19:35 03/20/87 c5STc.Rf] SECTION ADVECTICNS 5 7 9 11 12 S******** S******** S ******** * e.^5 * 0.073 * 0.134 * 0.019 * 0.099 * * ********** ********** * * S3ST * >>>>> * WFST * »»> * WEST * ********* ********** ********** 0.n23 0.183 *0.000 0.124 *0.Ô35 UP UP *** UP DOWN $*******£ ********* ********** *s******** * 0.049 * 0.034 * 0.016 * 0.026 * 0.283 * 0.304 * 0.146 * * ********** ********** ********** * * ^UST * >»>> * WEST * »»> * WEST * <<<<< * EAST * ********** ********* ********* ********* 0.T66* 0.001 0.091 0.078 0.124 **UP OOWN UP UP UP ********E ********* ********* s******** * 0.161 * 0.092 * 0.041 * 0.113 * 0.003 * 0.177 * 0.512 * 0.374 * 0.779 * * ********** ********** ********** ********** * * EAST * >>>>> * EAST * »»> * EAST * »»> * WEST * <«« * EAST * ********* ********* ********* ********* ********** 0.005 0.031 0.152 0.039 0.392 «0.674 DOWN OCWN UP UP UP OOWN S******** s******** ********£ ********* ********* ********* *s*******E * * 0.025 * 0.113 * 0.021 * 0.119 * 0.072 * 0.105 * 0.116 * 0.286 * 0.247 $ 0.504 * 0.057 * * * ********** ********** ********** ********** ********** ********** * * * <<<<< * EAST * >>>>> * EAST * >>>» * EAST * »»> * WEST * <<<« & EAST * <<<« * * ********* ********** ********* ********* ********* ********* ********* 0.012 0.000* 0.073 0.113 0.090 0.039 0.078 0.971 OOWN' *** OOWN OOWN UP OOWN UP UP S********* s******** ********£ ********* ********* ********* s*****ft*E * * 0.105 * 0.349 * 0.003 * 0.116 * 0.065 * 0.265 * 0.046 * 0.259 * 0.189 * 0.341 * 0.001 * * * ********** ********** ********** ********** ********** ********** * * * <<<<< * EAST * <<<<< * EAST * >»» * EAST * »»> * WEST * <<<<< * EAST * <<«< * * ********* ********* ********* ********* ********* ********* ********* 0.102 0.320 0.176 0.194 0.014 0.075 0.790 OOWN OOWN OOWN OOWN OOWN OOWN UP S******** ********* jj,***^}.^ ********* ********* ********* $*******£ * & 0.013 * 0.181 * 0.044 * 0.133 * 0.377 * 0.228 * 0.469 * 0.355 * 0.076 * 0.157 * 0.075 * * * *****5,«**** «<« * EAST ********** >>>>> ** WEST ********** »»> ** EAST ********** >»» ** WEST ********** »>» ** WEST ********** »>» ** ** ********* ********* ********* ********* ********* ********* ********* 0.000 0.443 0.278 0.330 0.052 0.031 0.357 UP OOWN UP DOWN DOWN UP UP ********* ********* ********* ********* ********* ********* ********* * * 0.000 * 9.620 * 0.135 * 0.341 * 0.022 * 0.029 * 0.185 * 0.065 * 0.466 * 0.123 * 0.065 * * * ********** ********** ********** ********** ********** ********** * * * * WEST * <<«< * EAST * >»>> * WEST * >»» * EAST * >»» * WEST * »>» * * ********* ********* ********* ********* ********* ********* ********* *&******** 0.00DOW0N *********** 0.000 ********** 0.040.00UP0 1 *********** <«<0.04< 1 ********** 0.090.00U»40 *********** 0.13<<<<<5 ********* 0.130.16DOWEASTN80 *********** »»0.16>3 ********* 0.160.00UP0 3 *********** 0.000 *********$* 0.190.00DOW0N 6 **&********* 0.19»»>6 ****************** 0.33UP 8 ** FOZ : GEQWTIGhT 4 19:19:35 03/20/87 WESTERN SECTION TURBULENCES 1 2 4 6 8 10 12 1********0.357 1*******1 0.075 i*******| * 0.000 ********** 0.000 ********** 0.000 * * * * * * * ********* ********* ********** 0.000 0.063 0.007 *0.114 * * * *** S*******E ********P s******** *s******** * * 0.025 * * 0.000 * * 0.229 * * * 0.000 ********** 0.127 ********** 0.000 ********** 0.000 * ** ** ** ** ********** ********* ********* ********* 0.000* 0.115 0.000 0.004 0.167 *** * * * * S*******E* ********E ********E ********* s******** * * 0.216 * * 0.055 * * 0.000 * * 0.577 * * * 0.000 ********** 0.000 ********** 0.095 ********** 0.311 ********** 0.409 * * * ** ** ** ** ********* ********* ********* ********* ********** 0.000 0.020 0.000 0'00° 0.000 *0.000 S******** s******** ********E ********E ********* ********* *s*******E * * 0.035 * * 0.131 * * 0.012 * * 0.000 * * 0.177 * * 0.186 * * * ********** 0.034 ********** 0.015 ********** 0.151 ********** 0.000 ********** 0.000 ********** * ** ** *« ** ** ** ** ********* ********* ********* ********* ********* ********* ********* 0.032 °»20^ !3*229 0.000 0.058 °«£23 1.869 S******** s******** ********ç ********£ ********* ********* s*******£ * * 0.054 * * 0.100 * * 0.024 * * 0.000 * * 0.107 * * 0.691 * * * ********** n.060 ********** 0.077 ********** 0.196 ********** 0.000 ********** 0.000 ********** * ** ** ** ** ** ** ** ********* ********* ********* ********* ********* ********* ********* 0.000 0.00" 0.000 0.000 0.000 0.252 0.397 ******* S******** ********* ********* ********* ********* ********* $*******£ * * 0.016 * * 0.031 * * 0.244 * * 0.039 * * 0.000 * * 0.093 * * * ********** 0.119 ********** 0.0 50 ********** 0.484 ********** 0.000 ********** 0.000 ********** * * * * * * * * * * * * * * * ********* ********* ********* ********* ********* ********* ********* S.080 H.272 0.017 0.000 0.000 0.014 0.172 ******* ********* ********* ********* ********* ********* ********* ********* * * O.OOO * * 0.000 * * 0.000 * * 0.147 * * 0.000 * * 0.000 * * * ********** 0.000 ********** Ç.106 ********** 0.161 ********** 0.000 ********** 0.000 ********** * ** * * ** ** ** ** ** ********* ********* ********* ********* ********* ********* ********* 0.038 0.044 0.000 0.010 0.000 0.000 0.000 ******* ********* ********* ********* ********* ********* ********* ********* * * 0.000 * * 0.053 * * O.OOO * * 0.232 * * 0.064 * * 0.000 * * * ********** 0.000 ********** 0.000 ********** 0.000 ********** 0.000 ********** 0.000 ********** * ** ** ** ** ** ** ** ********* ********* ********* ********* ********* ********* r-'".: : •-:-•)'A "I'J H r i 19:19:35 J3/20/37 cAST'c^N S = CTI2N TUR3ULEMC5S 1 ->, 5 ? 9 11 12 S******** S******** s******** * * 0.049 * * 0.000 * * * 0.010 ********** 0.000 ********** 0.000 * * * * * * * ********* ********** ********** 0.421 0.293 *0.000 0.319 *0.000 * * *** * *** S*******5 ********* ********** *$******** * * 0.000 * * 0.073 * * 0.209 * * * O.OOQ ********** 0.12? ********** 0.000 ********** 0.000 * ** ** ** ** ********** ********* ********* ********* 0.099* 0.000 0.082 0.000 0.374 *** * * * * S********* ********£ ********* ********* s******** * * 0.02 4 * * 0.000 * *0.241* *0.0 58* * * 0.000 ********** 0.000 ********** 0.095 ********** 0.311 ********** C.409 * ** * * ** ** ** ********* ********* ********* ********* ********** 0.15? 0.018 0.000 0.020 0.201 *0.133 S ******** s******** ********= ********* ********* ********* *s*******£ * * 0.000 * * 0.010 * * 0.039 * * 0.000 * * 0.197 * * 0.119 * * * ********** 0.034 ********** 0.015 ********** 0.151 ********** 0.000 ********** 0.000 ********** * ** ** $ * ** ** ** ** ********* ********** ********* ********* ********* ********* ********* 0.032 0.000* 0.000 C.000 0.000 0.012 0.000 1.869 * *** ****** S********* S******** ********C ********* ********* ********* $#******£ * * 0.000 * * 0.000 * * 0.017 * * 0.000 * * 0.110 * * 3.400 * * * ********** 0.060 ********** 0.077 ********** 0.196 ********** 0.000 ********** 0.000 ********** * ** ** ** ** ** * * * * ********* ********* ********* ********* ********* ********* ********* 0.000 0.000 0.100 0.000 0.052 0.135 0.397 ******* S******** ********* ********? ********* ********* ********* 5*****$*£ * * 0.000 * * 0.059 * * 0.000 * * 0.136 * * 0.071 * * 0.000 * * * ********** 0.119 ********** 0.050 ********** 0.484 ********** 0.000 ********** 0.000 ********** * ** ** ** ** ** ** * * ********* ********* ********* ********* ********* ********* ********* 0.080 0.000 0.150 0.000 0.137 0.Ö77 0.172 ******* ********* ********* ********* ********* ********* ********* ********* * * 0.000 * * 0.000 * * 0.000 * * 0.000 * * 0.101 * * 0.180 * * * ********** 0.000 ********** 0.106 ********** 0.161 ********** 0.000 ********** 0.000 ********** * ** ** ** ** ** ** * * ********* ********* ********* ********* ********* ********* ********* 0.038 0.000 0.071 0.213 0.058 0.000 0.000 ******* ********* ********* ********* ********* ********* ********* ********* * * 0.000 * * 0.003 * * 0.000 * * 0.000 * * 0.000 * * 0.009 * * * ********** g.000 ********** 0.000 ********** 0.000 ********** 0.000 ********** 0.000 ********** * ** ** ** ** ** ** ** ********* ********* ********* ********* ********* ********* ********* =02 t GEOWEIGFT 4 19:19:35 03/20/87 TOTAL PRODUCTIF "G 3.94375 ORG./INQRG. RATIO 5.68121 ORG. WESTERN SECTION DETRITUS FLUX INORG. (FLUXES MULTIPLIED 3Y TEN) 4 6 8 10 12 •i** ••>* A* *r *i* "c i* C S******** * -0.051* * -0.102* * * * * * * * -0.113* * -0.009* * -0.062* ********* ********* -,» Jp .,,•» «y» -,* -,*. -,* C S******** * -0.097* ï**SÎÎ0ÎÏ * 0.093* î*ïîîî!S * 0.044* * * * * * 0.009* * 0.000* ********* ********* * -0.015* ********* ********* T T" TT* "T* t* *V *** nf* *ï* S*******E ********£ ********* S******** * -0.058* * 0.000* * 0.048* * 0.090* * -0.063* * * * * * * * * * 0.000* * * * -0.063* * -0.0 44* ********* * 0.020* ********* ********* * 0.000* ********* ********* s ******** S*******A ********!= ********? ********* ********* * -0.0 6 7* * 0.0 3 5* * 0.077* * 0,000* * 0.000* * 0.223* * -0.780* * * * * * * * 0.00 0* * -0.O02* * * * * * 0.000* * * * 0.000* ********* * 0.008* * 0.000* ********* * 0.036* ********* ********* ********* ********* ********* s ******** S*A****** ********£ ********? ********* ********* S*******E * 0.066* * -0.024ft * 0.0 2 9* * o.ooo* * 0.000* * 0.151* * 0.674* * * * & * * * * * 0.000* * * * * * -0.Q02* * 0.0 04* * O.roB* * 0.0 00* ********* * 0.0 04* * 0.000* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* S*******E * o.ooo* * 0.046* * 0.021* * 0.000* * 0.000* * 0.006* * 0.0 5 9* * * * * * * * * * * * * * 0.000* * * * -0.003* * 0.001* * 0.0 42* * 0.009* * 0.000* ********* * 0.019* ********* s?******** ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* * 0.001* * 0.0**Ö* * 0.^06* * 0.003* * 0.031* * 0.0 36* * * * * * * * * * 3.000* * 0.000A * 0.019* * 0.000* * 0.001* * * * * ********* * 0.0 00* * 0.003* ********* ********* ********* ********* ********* ********* ********* ********* ********* vi******** ********* ********* * 0.014* * 0.001* * 0.001* * 0.007* * 0.006* * 0.007* * 0.012* * * * * * * * * * 0.026* * * * 0.056* * * * * * 0.000* * o.ooo* ********* * 0.000* * 0.0 04* ********* ********* ********* ********* * 0.000* ********* G=OW^TGHT 4 19:19:35 03/20/37 7JT4L PRODUCTION ?G î.^4975 QRG./I?1QRG. CM-5Q Shaw, R.W.s Estimated contributions- of Swedish and outside sources to background aerosol concentrations in Sweden. February 1981 , 31 pp. CM-51 Björkström, A.î On the inadequacy of one-dimensional ocean models -for the global carbon cycle. February 1981, 25 pp. CM-52 Bolin, B.: Changing global biogeochemistry. February 1981, 33 pp. CM-53 Shaw, R.W. and Rodhe, H.: Non-photochemical oxidation of SO2 in regionally polluted air during winter. March 1981 , 2? pp. CM-54 Söderlund, R. and Granat, L: Sodium CM-56 Söder lund, R. and Granat, L.: Chloride CM-59 Söderlund, R. and Granat, L.: Ammonium (NH^+> in precipitation - A presentation of data from the European Air Chemistry Network. July 1982, 6? pp. CM-60 Hamrud, M.: Residence time and spatial variability for gases in the atmosphere. November 1982, 6? pp. -61 Johansson, C, Rich ter, A. and Granat, L.: A system -for measuring -fluxes o-f trace gases to and from soil and vegetation with a chamber technique. September 1983, 15 pp. -62 Morales, C. and Rodhe, H.: Atmospheric visibility in the Scandinavian mountains - is there a secular- trend? December 1982, 15 pp. -63 Ross, H.: An automated method for the determination o-f Na, K, Mg and Ca using atomic absorption spectroscopy. June 1983, 14 pp. -64 Rodhe, H., Granat, L. and Söderlund, R.: Sul-fate in precipitation - A presentation o-f data from the European Air Chemistry Network. January 1984, 72 pp. -65 Hamrud, M.: Lagrangian time scales connected with clouds and precipitation. February 1984, 17 pp. -66 Ross, H.: Atmospheric selenium. March 1984, 68 pp. -67 Ross, H.: Methodology for- the collection and analysis of trace metals in atmospheric précipitât!on. November 1984. -68 Rodhe, H. and Hamrud, M.: On the design of a global detection system for airborne radioactivity. January 1985, 25 pp. -69 Söderlund, R., Granat, L. and Rodhe, H.: Nitrate in precipitation - A presentation of data from the European Air Chaemistry Network. October 1985. -70 Morales, C, Husar, R.B. and El Ghazzaway, 0.: Use of visibility observations for the investigation of hazy air masses. October 1986. -71 Bolin, B., Björkström, A., Holmen, K. and B. Moore: On inverse methods for combining chemical and physical océanographie data: a steady-state analysis of the Atlantic Ocean. July 1937. Abstract An attempt has been made to increase the spatial resolution in the use of inverse methods to deduce rates of water circulation and detritus formation by the simultaneous use of tracer data and the condition of quasi-geostrophic flow. It is shown that an overdetermined system of equations is desirable to permit analysis of the sensitivity of a solution to errors in the data fields. The method has been tested for the Atlantic Ocean, in which case we employ an 84-box model (eight layers in the vertical and 12 regions in the horizontal define the box configuration). As quasi-steady tracers we consider dissolved inorganic carbon (DIC), radiocarbon, alkalinity, phosphorus, oxygen, salinity and enthalpy. The condition of quasi-gecistrophic flow is employed for flow across all vertical surfaces between regions except close to the equator. Water continuity is required to be exactly satisfied by the use of a set of closed loops to describe the advective flow. The method of least squares is used to derive a solution and in in so doing we also require, that turbulence only transfers matter down tracer gradients (i.e. eddy diffusivity is nonnegative) and that detritus is formed only in surface boxes and is destroyed in the water column below. It is shown how appropriate weighting of the equations in the set is decisive for the solutions that we derive and that great care must be taken to ascertain that the interior tracer distributions and the boundary conditions in terms of exchange of tracer material with the exterior are compatible. The enthalpy equations turn out not to fulfil such a demand and have not been used in deriving the solutions presented in the paper. A basic solution with an ageostrophic flow component of about 15% is derived and compared with current knowledge about the general circulation of the Atlantic Ocean and the rates of detritus formation and destruction. The sensitivity of the solution to uncertainties in the data field is presented. It is shown that the solution is markedly dependant on whether turbulent transfer is included or not. On the other hand, we are not able to determine how Important boundary conditions on exchange of C02 and C between the atmosphere and the sea are for our solution and accordingly for the uptake of present excess in the atmosphere. Nor is it possible to determine with any degree of confidence which of different sets of Redfield ratios would best satisfy the present models for detritus formation and destruction. Also the most likely rate of deep water overturning cannot be determined very accurately. The uncertainty of current estimates found in the literature is also considerable. Some of the difficulties arise from a mismatch between the questions asked and the resolution of the data; an example is the turnover time for individual deep ocean boxes versus the small gradients in the corresponding 141C data. In other cases, the structure of the model with its large boxes does not take full advantage of the information content in the data. We conclude with some observations about future research efforts that might successfully address these difficulties. */ Institute for the Study of Earth, Oceans and Space, University of New Hampshire Durham, New Hampshire 03824 USA and Laboratoire de Physique et Chimie Marines, Université Pierre et Marie Curie, Paris VI, France. 1 Performing organization University of Stockholm' DKPAKTMEN'JT 01» MCTE0R0L00Y . 8 Project: 9 MI Report number jl Planning (2 Amplifie 5fcion | 13 Concluded 10 11 Contract number 12 Starting year 13 Finishing yoar 14 MI Report number 15 Sponsoring organization Swedish Natural Science Research Council under Contracts E-EG 0223-112, E-EG 113, E-EG 0223-115 and E-EG 0223-118. US DOE under Contract DE-AC05-84OR214OQ. 16 Title and subtitle of project or report On inverse methods for combining chemical andphysical océanographie data: A steady state analysis of the Atlantic Ocean. 17 Project leader/Auther(s) Bert Bolin, Anders Björkström., Kim Holmen and Berrien Moore* 18 Abstract (goal, method, technique,result'etc.) Please turn over 19 Abstract written by Authors 20 Key words Tracers; Ocean circulation; Carbon cycle; Ocean chemistry; Matrix inversion; Constrained minimization. 21 Classification system and class(es) UDC 551.464:551.465 22 index terms (source) 23 Bibliographical data 24 ISSN University of Stockholm, Department of Meteorology, 0280-445X Report CM-71 25 ISBN 26 Secret I paragraph 27 Language 28 Number of pages 29 Price SiOfficial Secrets Act English 220 ,30 Distribution by Library, Department of Meteorology University of Stockholm Arrhenius Laboratory S-106 91 STOCKHOLM Sweden