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.