Quantifying Dynamical Proxy Potential through Oceanic Teleconnections in the North Atlantic

Oceanic quantities of interest (QoIs), e.g., ocean heat content or transports, are often inaccessible to direct observation, due to the high cost of instrument deployment and logistical challenges....

timates remain uncertain, due to a sparse array of current meter moorings and the sen- ing the ISR (Fig. 2) and are therefore not expected to be ideal placements for monitor- 148 ing ISR heat transport. We will show that these observations nevertheless provide par-149 tial constraints on the QoI through shared adjustment physics, which are uncovered and 150 quantified by dynamical proxy potential. 151 Here, we work within the ECCO (Estimating the Circulation and Climate of the NAP measurements are available. We note that the quantification of dynamical proxy 155 potential does not require actual (here: OSNAP and OVIDE) observational data, since 156 it investigates dynamical relationships in the model equations, rather than observed co-contrast, the model state variables within the white box in Fig. 1(c), e.g., temperature 173 and velocity, adjust freely following the model dynamics, to ensure dynamic and kine-174 matic consistency. An implicit assumption in ocean state estimation is that the control 175 variables F m (x i , y j , t k ) comprise all possible sources of changes in the ocean state and 176 circulation and that the model is perfect (Tarantola, 2005;Wunsch, 1996).
198 2. projecting Q onto V, and 199 3. taking the square of this projection.

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These three steps result in the dynamical proxy potential of the observed quantity for 201 the QoI: tity (pink box) for an unobserved QoI (purple box) measures 217 (i) the similarity between the ocean adjustment physics for the observed vs. unob-218 served quantity (pink vs. purple arrows) in response to changes in forcing (green 219 box), on a scale from 0% (no similarity) to 100% (identical ); 220 (ii) the relative uncertainty reduction in the QoI that would be achieved if the obser-221 vation were to be added without noise to the state estimation framework in Fig. 1(c).

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The flow of information and uncertainty reduction within the state estimation frame-223 work -from the observation via the controls to the QoI -is delineated by the black 224 arrows in Fig. 1  The QoI in our case study is heat transport across the Iceland-Scotland ridge, de-256 noted by HT ISR . We investigate three different temperature observations in the North 257 Atlantic, located inside the green dots in Fig. 2 and labelled by θ A , θ B , and θ C . Obser-258 vations θ A and θ C are located in the Irminger Sea at (40 • W, 60 • N), while observation 259 θ B is situated in the eastern North Atlantic off the Portuguese coast at (12 • W, 41 • N). 260 θ A and θ B are subsurface observations, situated at 300 m depth, and θ C is a surface ob-261 servation. 262 We quantify the dynamical proxy potential of the five-year mean of the observa- COv4r2 state estimate. Dependence of the specific evaluation period is weak, given that 266 HT ISR , θ A , θ B , and θ C depend approximately linear on the forcing variables in Table 1 267 (Appendix A).

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The QoI, as simulated by the model, is diagnosed as follows:    Meridional wind stress τ y five years 0.05 N/m 2 plies that the sensitivity projection (eq. (5)) for each individual forcing field is fully determined by the adjustment physics, and not by the forcing weight. In contrast, spatially    Table 1. The normalization factor, σ HT = 11 TW, is computed according to eq. (3).

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The bar chart in Fig. 3(e) shows the relative importance of the four forcings F m for im-314 pacting HT ISR . Relative importance is measured by the ratios  Information required to recover HT ISR ative to the remaining forcing fields, is surprisingly small: only 3% (Fig. 3(e)). We note 349 that even if we tripled ∆Q net in Table 1, while keeping the weights for the remaining forc-350 ings unchanged, HT ISR would still be less sensitive to Q net,↑ than to any of the remain-351 ing three forcing fields in Fig. 3  Information captured by observation The large-scale wind stress sensitivity patterns of θ C (Fig. 4(f)) are very similar 388 to the ones of θ A (Fig. 4(d)), except that they are of much weaker amplitude. The sim-389 ilarity of the patterns suggests that the surface observation θ C is sensitive to similar re- pattern that is seen in the northeast Atlantic in Fig. 5(a). The anomalous warming in-  (Figs. 4(d),(f)). We note that, in this region, the pattern of τ y 420 sensitivities for θ A , θ C (Figs. 4(d),(f)) are comparable to those for HT ISR (Fig. 3(d)), 421 except that sensitivities of θ A , θ C are of opposite sign, compared to HT ISR (see Figs. 6(c),(d) 422 for a side-by-side comparison).

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To explain the opposite signs, we perform a second perturbation experiment sim-424 ilar to the one presented in Fig. 5(a). In the second experiment, we increase northward 425 wind stress along the western Icelandic coast, in the region highlighted in Fig. 5(b), where 426 θ A and θ B show positive sensitivity (Fig. 4(d),(f)) and HT ISR shows negative sensitiv-427 ity (Fig. 3(d)). Fig. 5(b) shows the response anomaly in subsurface temperature, at a  Fig. 2). The weak-437 ened northward transport of warm Atlantic waters across the ISR leads to the anoma-438 lous cold temperatures that are seen in the Norwegian Sea in Fig. 5(b), and is consis-439 tent with a reduced HT ISR , as predicted by the negative sensitivities in Fig. 3(d). The  The perturbation experiment presented in Fig. 5(b) explains the opposite sign in 445 the sensitivities along the western Icelandic coast in Fig. 3(d) vs. Fig. 4(d). The fact that  Fig. 5(b).   Information required to recover the QoI, cross-ridge heat transport HT ISR Information captured by the observation, subsurface temperature θ A Fig. 3(a) (a) σ −1 HT ∂(HTISR) ∂Q net,↑ ∆Q net,↑ Fig. 4(a)  HT ISR is relative insensitive to Q net,↑ and E-P-R ( Fig. 7(a)). Note that even for the sur-481 face temperature observation θ C , which is highly sensitive to surface heat fluxes (Fig. 7(d)), 482 the Q net,↑ contribution to the projection (11) is negligible (Fig. 7(k)).  (Fig. 7(e)) and θ A (Fig. 7(f)), 489 as discussed in section 3.3 and Fig. 5(b). The negative projection in the northeast At-490 lantic exceeds the positive projection in the eastern Atlantic waveguide (Fig. 7(i)). To-491 tal positive and negative contributions sum to Q•V A = −0.44 (Fig. 7(i)). Here, par- Positive wind stress contributions to Q•V B (Fig. 7(j)) are of similar magnitude 501 as positive wind stress contributions to Q•V A (Fig. 7(i)), due to the pressure adjust-502 ment mechanism in the eastern Atlantic waveguide, shared among θ B , HT ISR (and θ A ).

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As a result, θ A (θ B , θ C ) captures 19% (1%, 1%) of the variability of HT ISR , taking into 517 account all potential forcing scenarios. As for (ii), dynamical proxy potential predicts 518 that uncertainty in HT ISR would get reduced by 19% (1%, 1%), if a noise-free measure-519 ment value of θ A (θ B , θ C ) was added to the state estimation framework that was described 520 in section 3.1. ties. We note, however, that the concept is readily generalized to cases where all of the 537 restrictions are relaxed, and we will present such generalized applications in forthcom-538 ing work. In the following, we summarize the results from our exemplifying case study, 539 point out potential limitations, and provide specific directions for future work.

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In this work, heat transport across the Iceland-Scotland ridge (HT ISR ) was the ex-541 emplary QoI. We explored the potential for three example observed quantities to serve 542 as proxies for this QoI: two temperature observations, θ A , θ C , in the Irminger Sea, and sidering the five-year mean of the two quantities, we find that the dynamical proxy po-562 tential of θ A for HT ISR is 19%. Dynamical proxy potential allows two equivalent inter- ing dynamical constraints contained within the existing observational database.

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(2) The information that the observation θ A provides for the QoI HT ISR originates 577 in the observation's potential to capture a certain degree of the variability of HT ISR 578 -a similar concept that underlies statistical proxy potential. However, in contrast 579 to statistical proxy potential, dynamical proxy potential only accounts for covari-580 ability that has a dynamical underpinning, by employing sensitivity information 581 that traces variability of HT ISR and θ A back to its origins in local or remote forc- plementary information from further observations. In our case study, we found that 611 θ B is sensitive to wind forcing in region (I), but entirely insensitive to wind forc-612 ing in region (II). A forthcoming paper will show that considering θ A and θ B in 613 combination, will help to extract some of the information which is lost in destruc-614 tive interference when viewing θ A in isolation.
plete sensitivity information to local and remote forcing. This sensitivity information, 617 in turn, is the cornerstone of the quantification of dynamical proxy potential, laying the 618 groundwork for dynamics-based observing system design. In previous work, adjoint-derived 619 sensitivity information has been proposed to support observing system design in a dis-