Error Budget of Inverse Box Models: the North Atlantic
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NOVEMBER 2003 GANACHAUD 1641 Error Budget of Inverse Box Models: The North Atlantic ALEXANDRE GANACHAUD Institut de Recherche pour le DeÂveloppement, Laboratoire d'Etudes Geophysiques et d'Oceanographie Spatiale±UMR5566/CNRS/CNES/ IRD/UPS, NoumeÂa, New Caledonia (Manuscript received 29 March 2002, in ®nal form 21 March 2003) ABSTRACT Linear inverse box models based on hydrographic data are widely used to estimate the ocean circulation and associated transports of heat and of other important quantities. The inverse method permits calculation of a circulation that is consistent with basic conservation laws such as those for mass or salt along with uncertainties. Both uncertainties and solution depend upon assumptions about noise and the limitations of linear models. Internal waves introduce noise into the measurements, while ocean variability sets bounds on the linear model skills when time average circulation is sought. Observations of internal wave spectra and sensitivity experiments on hydrographic data suggest estimates for measurement errors in transport calculations of 63Sv(1Sv8 106 m3 s21 . 109 kg s21) at midlatitudes, with dependence on latitude. The output of a realistic numerical ocean model is used to quantify the impact of ocean variability on one-time hydrographic sections. The implied error is of order 68 Sv in net mass balance and 6100 kmol s21 in net silica balance. (The estimates are a priori errors.) Net mass balance has often been assumed more accurate with errors in individual layers compensating. While this may be true instantaneously, it is shown here that such an assumption leads to incorrect estimation of the time mean transports and diapycnal transfers. 1. Introduction heat, and salt conservation. Modi®cations are made to the reference velocities (and any other adjustable pa- So-called ``inverse box models,'' ®rst described by rameters) so as to force the circulation to consistency Wunsch (1978), are now commonly used to estimate the with the unsatis®ed requirements. ocean circulation and transports using hydrographic data However, because of the existence of noise and in- (e.g., Macdonald and Wunsch 1996; Robbins and Toole 1997; Tsimplis et al. 1998; Ganachaud and Wunsch complete physics in the geostrophic model, no con- 2000). These models provide a formal link among data, straint should be satis®ed exactly. The solution depends ocean dynamics, and a priori knowledge of the circu- directly upon the degree to which the constraints are lation. Sets of temperature, salinity, and sometimes ox- satis®ed, and one must attempt to be as realistic as pos- ygen and nutrient measurements at depth and across a sible in estimating the expected error in any require- transoceanic section are used to obtain a best estimate ment. The purpose of this paper is to reexamine from of the velocity ®eld and property transports, along with the beginning the question of noise in hydrography, and an estimate of their respective uncertainties. From tem- to provide zero-order, but nonetheless fully quantitative, perature and salinity measurements, the geostrophic values. Mass conservation is the most basic, and usually ¯ow (thermal wind) is calculated relative to a given the most important, of all imposed constraints; it is also reference surface. This ``dynamic method'' usually em- representative of other conservation requirements. We ploys the reference surface as one where the initial ve- also brie¯y examine tracer conservation constraints, us- locity estimate is zero. In practice, any additional in- ing silicate as the example. formation, such as that from current meters, can be em- Few estimates of prior uncertainties are available. ployed to establish the initial velocity at the reference Roemmich (1980) estimated that total mass could be surface (``reference velocity''). The thermal wind in- conserved within 61to62Sv(1Sv5 10 6 m 3 s21) tegrated from that reference gives an initial estimate of between two trans-Atlantic sections without incurring the general circulation, but one which usually fails to large changes to the initial circulation. Several studies satisfy basic requirements (``constraints'') such as mass, ``tested'' the ability of inverse models to reproduce a numerical ocean circulation with various inverse models (Zhang and Hogg 1996), with success depending upon Corresponding author address: Dr. Alexandre Ganachaud, Institut both model and constraints. For instance, McIntosh and de Recherche pour le DeÂveloppement, Centre de NoumeÂa, NoumeÂa BP A5, New Caledonia. Rintoul (1997) applied the method to an instantaneous E-mail: [email protected] snapshot of the Southern Ocean circulation from a qua- q 2003 American Meteorological Society Unauthenticated | Downloaded 10/02/21 12:14 AM UTC 1642 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20 si-eddy-resolving general circulation model (GCM). East Top(x) They found that the inverse calculation was successful T 5 EEdx dz r(x, z)C(x, z) as long as the prior statistics of the solution correspond- West Bot(x) ed to the correct (i.e., GCM) solution, as linear esti- 3 [b(x) 1 y R(x, z)] mation theory shows must be true. They did not include any measurement noise and constrained mass to be con- 5 TbR1 T , (2) served within 60.01 Sv which would be unrealistic for where C is the tracer concentration per unit mass (C 5 real data. 1 for mass), Tb is the reference velocity contribution, The full inverse problem for a box model is a non- and TR is the contribution from relative velocities. For linear one (Mercier 1986; Wunsch 1994) because one individual oceanic layers, Top(x) and Bot(x) de®ne the could permit modi®cation of the underlying temperature upper and lower boundaries of the layer. Steady-state and salinity ®elds to correct for time changes or mea- mass conservation may be written surement errors. By ®xing the temperature and salinity, =´(rv) 5 FW, the problem is linearized, but one introduces a residual error into the calculation. This error is estimated here where v is the three-dimensional velocity vector and in the North Atlantic. FW is the net freshwater ¯ux into the area (hereafter, Section 2 presents the formalism of the problem and FW 5 0 for simplicity). The corresponding equation basic notations; section 3 gives estimates of the effect for tracers is of oceanic variability on one-time sections; section 4 =´(rC) 5 =´k=(rC) 1 Q, (3) quanti®es the various measurement noise sources; sec- tion 5 provides values for the a priori size of the solution where k is an eddy diffusion tensor and Q is a possible (reference velocities); while section 6 analyzes the con- source or sink. It is assumed that the spatial sampling servation of silicate, followed by a general discussion along a section is ®ne enough to resolve most of the and summary of the error budget. horizontal eddy tracer transport; that is, smaller-scale lateral processes including horizontal diffusion are con- tained in the noise. Integrated over a layer bounded 2. Formalism horizontally by zonal hydrographic sections and land, and vertically by isopycnal surfaces, and rearranged, (3) Consider a zonal hydrographic section. The geo- becomes (appendix A) strophic velocity ®eld between pairs of stations is de- axT 1 n 5 y (4) rived from thermal wind, with y aR(x, z) 5 y (x, z) 1 b(x) {b } ii51,Np z g ]r x 5 {w*}kk51,N 21 , 52 dz 1 b(x), (1) l {k*} r 0 f E ]x kk51,Nl21 z 0(x) where z is the vertical coordinate (increasing upward); where x contains the unknowns bi at each station pair and diapycnal transfers {w*kk } and {k* }, and x can in- x is the horizontal coordinate along the section; z 0(x)is an arbitrary reference depth; y (x, z) is the absolute clude other parameters such as adjustments to Ekman a transport at the surface or freshwater transport; N is velocity, y (x, z) is the thermal wind relative to z (x) p R 0 the number of station pairs; N the number of layers and; and b(x) is the unknown reference velocity at z (x) (or l 0 a is the discrete integrals of T and of dianeutral trans- a correction to an optimal guess); g is gravity; and f is b fers. The term y contains the relative transport diver- the Coriolis parameter. The ``reference surface'' z 0(x) gence (TR) and the source term Q, including the Ekman is speci®ed at each pair according to prior knowledge. transport convergence. As (4) cannot be satis®ed exactly Both b and y are normal to the section and represent with real data, n represents the noise. Equations of the horizontal averages between the stations. The initial cir- form (4) for different layers and properties are grouped culation is inferred by elaborate guesses of the reference to form a single matrix equation, surface, z 0(x), at which b(x) would almost vanish. The inverse calculation provides an estimate of the adjust- Ex 1 n 5 y, (5) ment {bi} 5 b to this a priori solution, along with its where E is M equations 3N unknowns. Constraints on error covariance P. This solution satis®es the constraints net transports in particular regions (e.g., Florida Strait) on the circulation and is consistent with the a priori are added by choosing appropriate elements for E and variance of b. (Bold lowercase letters represent column y. Various methods have been used to solve (5) for x vectors, while bold uppercase letters are matrices.) and n, with corresponding solutions and uncertainties The net transport of a tracer C is a linear combination depending on the a priori knowledge that one has on of b and y R, their size. While the dependence is formally speci®ed Unauthenticated | Downloaded 10/02/21 12:14 AM UTC NOVEMBER 2003 GANACHAUD 1643 via the covariance matrices for methods such as Gauss± Markov (Wunsch 1996), the a priori knowledge will guide the author to choose a solution, that is, the rank for the singular value decomposition method.