# Dynamical attribution of oceanic prediction uncertainty in the North Atlantic: application to the design of optimal monitoring systems

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## Abstract

In this study, the relation between two approaches to assess the ocean predictability on interannual to decadal time scales is investigated. The first pragmatic approach consists of sampling the initial condition uncertainty and assess the predictability through the divergence of this ensemble in time. The second approach is provided by a theoretical framework to determine error growth by estimating optimal linear growing modes. In this paper, it is shown that under the assumption of linearized dynamics and normal distributions of the uncertainty, the exact quantitative spread of ensemble can be determined from the theoretical framework. This spread is at least an order of magnitude less expensive to compute than the approximate solution given by the pragmatic approach. This result is applied to a state-of-the-art Ocean General Circulation Model to assess the predictability in the North Atlantic of four typical oceanic metrics: the strength of the Atlantic Meridional Overturning Circulation (AMOC), the intensity of its heat transport, the two-dimensional spatially-averaged Sea Surface Temperature (SST) over the North Atlantic, and the three-dimensional spatially-averaged temperature in the North Atlantic. For all tested metrics, except for SST, \(\sim\) 75% of the total uncertainty on interannual time scales can be attributed to oceanic initial condition uncertainty rather than atmospheric stochastic forcing. The theoretical method also provide the sensitivity pattern to the initial condition uncertainty, allowing for targeted measurements to improve the skill of the prediction. It is suggested that a relatively small fleet of several autonomous underwater vehicles can reduce the uncertainty in AMOC strength prediction by 70% for 1–5 years lead times.

## 1 Introduction

Anthropogenic global warming has changed Earth’s climate over the last century (IPCC 2007, 2013) In this context, there is an ever increasing societal pressure to predict (as accurately as possible) climate changes from local to global scales and from seasonal to centennial time scales. Previous studies have suggested that on interannual to decadal time scales internal variability dominates the uncertainty in global temperature changes, whereas on multidecadal to centennial time scales the emission scenarios are more important (Hawkins and Sutton 2009; Branstator and Teng 2012; IPCC 2013). Because of its slow variability and its large heat content, the ocean can control climate variations on interannual to decadal time scales. For example, variability of the Atlantic Meridional Overturning Circulation (AMOC) can suppress anthropogenically forced warming in the North Atlantic (Drijfhout 2015). Hence predicting AMOC variations, and more generally the North Atlantic Ocean state is crucial for accurate climate predictions on interannual to decadal time scales. Moreover, because of its robust quasi-harmonic multi-decadal variation (Kushnir 1994; Frankcombe and Dijkstra 2009; Chylek et al. 2011), the North Atlantic is a perfect candidate for successful interannual to decadal climate prediction (Griffies and Bryan 1997).

Considering the chaotic nature of several components of the climate system, in particular the atmosphere (Lorenz 1963), a pragmatic approach has been widely used to determine climate predictability on different time scales (Hurrell et al. 2006; Meehl et al. 2009). This approach consists of acknowledging the uncertainty in initial conditions by taking an ensemble of nearby initial conditions and evaluating the predictability by computing the time evolution of the spread of this ensemble (e.g., Boer 2011). There are various strategies to accurately sample the initial condition uncertainty and to accurately define the ensemble (e.g., Persechino et al. 2013). However, except for a few dedicated studies (e.g., Du et al. 2012; Baehr and Piontek 2014; Germe et al. 2017b), the role of the oceanic uncertainties has often been overlooked in these ensemble design strategies, despite being a possible source of predictability (Hawkins et al. 2016).

Since the early work of Griffies and Bryan (1997), there has been a large body of work suggesting the predictability on interannual to decadal time scales of climatically relevant metrics of the North Atlantic Ocean, such as the AMOC (Collins and Sinha 2003; Pohlmann et al. 2004; Collins et al. 2006; Latif et al. 2006; Keenlyside et al. 2008; Msadek et al. 2010; Teng et al. 2011). For example, Branstator and Teng (2014) suggested predictability of up to 20 years for one AMOC mode of variability, whereas Hermanson and Sutton (2010) suggested an average predictability of 5 years for the annual AMOC. Although using different methods to assess predictability, these two examples illustrate the difficulty of a robust quantitative estimation of AMOC predictability. Hence, despite good qualitative progress in assessing the interannual to decadal predictability of the North Atlantic climate variability, the quantitative results still substantially differ between studies. This disagreements might be caused by the model uncertainty that has been showed to dominate on decadal time scale (Hawkins and Sutton 2009). Beyond this, one other hypothesis to explain the lack of quantitative agreement is the rather small number of members (\(\sim\) 10) used to build the ensembles (Sévellec and Sinha 2017). This number is far from being enough to systematically sample the initial condition uncertainty of the climate system (Deser et al. 2012). Hence another methodology is highly desirable to assess predictability on interannual to decadal time scales.

Generalized Stability Analysis has been developed to estimate the transient growth of small perturbations (GSA; Farrell and Ioannou 1996a, b) rather than their asymptotic growth, as used in classical Linear Stability Analysis (Strogatz 1994) This generalization to finite time development makes GSA particularly well suited to study predictability (Palmer 1999). Hence, in the context of the North Atlantic there has been a wide range of applications with this method, from idealized ocean models to fully coupled ones (Tziperman and Ioannou 2002; Zanna and Tziperman 2008; Alexander and Monahan 2009; Hawkins and Sutton 2011) and even observations (Zanna 2012). Sévellec et al. (2007) suggested a subtle but fundamental modification of the GSA: instead of estimating the maximum growth of perturbations through a quadratic norm, one can estimate the maximum change in any linear combination of the state-variables.

This modification simplifies the solution going from an eigenvalues problem (i.e., singular value decomposition, hard to tackle in a state-of-the-art climate model) to an explicit solution (i.e., Linear Optimal Perturbation—LOP, Sévellec et al. 2007). This allowed the study of a wide range of climatically relevant problems of the North Atlantic from idealized ocean models (Sévellec et al. 2007) to state-of-the-art ocean general circulation models (Sévellec and Fedorov 2017). Also, through this new formulation of GSA, changes are not restricted to non-normal growths and are easily comparable to the pragmatic ensemble approach based on perturbing initial conditions.

These methods are particularly useful to determine regions of sensitivities to ocean initial conditions (Montani et al. 1999; Leutbecher et al. 2002). They have been shown to be extremely efficient in improving typhoon forecasts (Qin and Mu 2011; Zhou and Mu 2011), for instance. Hence, in the context of the AMOC, Wunsch (2010) and Heimbach et al. (2011) suggested the use of these methods to assess regions where enhanced sampling strategy can improve prediction. Alternative methods to assess the effect of enhanced sampling strategies have also been proposed, such as initializing different layers of the ocean (Dunstone and Smith 2010) but these methods are not yet able to determine the most sensitive region with the same level of details.

In this study, the relation between the two approaches is discussed and it is shown that under two assumptions (linearized dynamics and normal distribution of uncertainties), the spread of an ensemble can be retrieved from a theoretical framework. Unlike the ensemble simulation strategy, the theoretical framework provides an exact quantitative estimate of the predictability (measured through the Predictive Power), as it does not depend on the arbitrary number of members like the ensemble method. In Sect. 2, we develop the method for a quantitative assessment of the predictability and illustrate the solution using an idealized stochastic model. In Sect. 3, we apply the solution to a state-of-the-art ocean general circulation model (GCM) for four ocean metrics related to the AMOC. Applications to the design of efficient monitoring systems for climate prediction are given in Sect. 4. A discussion on the limits of the methods, conclusions and directions for future work are included in Sect. 5.

## 2 Theory

### 2.1 Propagating errors in a linear framework

Here we discuss the mathematical framework of our method. Readers more interested in its application to the North Atlantic Ocean circulation can skip this section.

*t*is time. The state vector is comprised of the three dimensional fields of temperature, salinity, and zonal and meridional velocities, together with the two dimensional field of barotropic stream function. Since we study a finite-dimensional vector space, we can also define a dual vector \(\left\langle {{\varvec{U}}} \right|\) through the Euclidian scalar product \(\left\langle {{\varvec{U}}|{\varvec{U}}} \right\rangle\).

*t*induced by the initial disturbance at time \(t_i\), such that \(\left| {{\varvec{u}}(t_i)} \right\rangle\) = \(\left| {{\varvec{u}}^\mathrm{ini}(t_i)} \right\rangle ;\) \(\left| {{\varvec{u}}^\mathrm{sto}(t)} \right\rangle\) is the perturbation induced by the stochastic forcing; and \({{\mathsf {{M}}}}(t,t_i)\) is called the propagator of the linearized dynamics from the initial time \(t_i\) to a time

*t*. In general the propagator does not commute with its adjoint, \({{\mathsf {{M}}}}^\dagger (t_i,t){{\mathsf {{M}}}}(t,t_i) \ne {{\mathsf {{M}}}}(t,t_i){{\mathsf {{M}}}}^\dagger (t_i,t)\), in which case the dynamics is said to be non-normal.

### 2.2 Ensemble spread and predictability

To evaluate the ocean state predictability we will focus on physical metrics of the form \(\left\langle {{\varvec{F}}|{\varvec{u}}} \right\rangle\), where \(\left| {{\varvec{F}}} \right\rangle\) is a vector defining the cost function. The derivation below is adapted from the theoretical framework set by Chang et al. (2004) to fit the specific goals of our analysis.

*N*initial oceanic disturbances (\(\left| {{\varvec{u}}^\mathrm{ini}_k(t_i)} \right\rangle\)) and of corresponding surface atmospheric disturbances (\(\left| {{\varvec{f}}^\mathrm{sto}_k(s)} \right\rangle\)), respectively (

*k*being the index of the random realization or of the individual member in the ensemble).

*N*initial and stochastic disturbances and project them onto the cost function or propagate (with the adjoint) the cost function only once and project it on the

*N*initial and stochastic disturbances. Obviously, the latter is a lot more efficient and will even allow to rebuild the entire probability density function (PDF). Indeed, the application of a moment (here the second moment or variance) is applied as a post-computation diagnostic. Hence, this demonstration can be reproduced for any statistical moment (not only the variance), so that the statistical properties of the oceanic initial condition and atmospheric stochastic forcing uncertainties can be propagated in the same way. Applying this procedure on any statistical moment of the PDF (\(\mu _n^\mathrm{ini}\) and \(\mu _n^\mathrm{sto}\)), we obtain:

### 2.3 Application to an idealized stochastic model

*k*is the index of the random realization or of the individual member of the ensemble, and \(\gamma\) is the inverse of an oceanic damping time scale set to 10 years.

## 3 Application to a detailed ocean GCM

### 3.1 Experimental set-up

To apply the exact same procedure as in Sect. 2.3, but now in a more realistic setting, we use a forced ocean GCM (NEMO-OPA 8.2, Madec et al. 1998) in a global realistic configurations (with a horizontal resolution of 2\(^\circ \times\) 2 \(^\circ\) and 31 vertical levels with a level distance ranging from 10 to 500 m, Madec and Imbard 1996). We also use its tangent and adjoint components (OPATAM, Weaver et al. 2003). The combined model configuration follows Sévellec and Fedorov (2017) under flux boundary condition. We refer the reader to this study for further details of the model configuration and the climatological background state used here.

Despite being the typical configuration used in current climate prediction systems and in the last Coupled Model Intercomparison Project (CMIP5, Taylor et al. 2012), the use of non-eddy-resolving model is not trivial. Indeed the ocean chaotic behaviour is absent from this low resolution model. However, the oceanic solutions of these type of models remain irregular through ocean–atmosphere interactions and the propagation of the atmospheric uncertainty in the (almost-laminar) ocean (Germe et al. 2017b). As a result, in this study we assume that the oceanic uncertainty is forced by the chaotic nature of the atmosphere.

We select four typical ocean metrics of the North Atlantic. The intensity of the AMOC (MVT, measured as the meridional volume transport above 1500 m at 50\(^\circ\)N), the meridional heat transport (MHT, measured at 25\(^\circ\)N), the spatially-mean SST (SST, average from 30\(^\circ\)N to 70\(^\circ\)N) and the oceanic heat content (OHC, measured as the mean temperature from 30\(^\circ\)N to 70\(^\circ\)N and from the surface to the bottom of the ocean).

### 3.2 Error growth attribution

*t*= 40 year for a given metric (Fig. 3c1–4). This diagnostic is 0 if the uncertainty reaches its asymptotic value and 1 if negligible (negative values suggest that the uncertainty exceed asymptotic value).

Our analysis shows that the uncertainty for the SST metric is almost equal to its asymptotic value after a few years (Fig. 3a3). Also, the oceanic initial condition uncertainty does not seem to play an important role for SST (Fig. 3b3). This suggests that the atmospheric synoptic noise is the main driver of the error growth. This leads to a rather weak Predictive Power over the 40 years tested (Fig. 3c3), suggesting that our ability to predict (i.e., potential prediction skill of) SST is restricted to values of less than 20% of its long-term variance and to interannual time scales. We emphasize, however, that this metric is not well represented in a forced ocean context. Hence conclusions from this experimental set-up might not be directly applicable to the coupled climate system.

For the three other metrics (MVT, MHT, and OHC) the uncertainty reaches its asymptotic value on much slower, multidecadal time scales (Fig. 3a1, 2, and 4). For OHC, however, the standard deviation is still slightly increasing at 40 years. This might be problematic since \(\sigma ^2(40~years)<\sigma ^2_\infty\), which potentially lead to an underestimation of the predictive power. Unlike SST, the OHC metric is also sensitive to the oceanic initial condition uncertainty. In the perfect case (absence of oceanic initial condition uncertainty), the error growth increases almost monotonically with time (Fig. 3a4), leading to an almost linear decrease of the Predictive Power from 1 to 0 over the 40 years tested (Fig. 3c4). This suggests that, in the absence of oceanic initial condition uncertainty, OHC has a predictive skill that remains above 80% up to a decade, above 50% up to 20 years, and below 20% after 3 decades. When not neglected the impact of oceanic initial condition uncertainty are mainly occurring over the first two decades (Fig. 3a4). Depending on the intensity of this error it can significantly impact the Predictive Power (Fig. 3c4). In the most extreme case it suggests the absence of potential prediction skill after only 5 years. In this case it induces a slight overshoot (a variance bigger than the asymptotic value) of the OHC variance peaking between 10 and 20 years. This implies that an initial error of such an intensity has huge repercussion on prediction systems by pushing them beyond their natural attractor. However, using the average value for the oceanic uncertainty, we find that it dominates the error growth on time scales up to 15 years, with a maximum impact of 75% of the error growth on interannual time scales (Fig. 3b4). This suggests that accurate oceanic initial condition can improve significantly the potential prediction skill of OHC on interannual to decadal time scales.

The two last metrics (MVT and MHT) show the same overall behaviours, but differ from OHC. In the absence of oceanic initial condition uncertainty, the two error growths increase with time until a saturation value is reached around 20–30 years (Fig. 3a1–2). This reflects on the predictive powers as an almost monotonous decrease in the exception of a plateau over the first \(\sim\) 5 years (Fig. 3c1–2). When an oceanic initial condition uncertainty is applied the instantaneous error growth of the two metrics becomes huge, regardless of the applied intensity (Fig. 3a1–2). This suggests that initial error on the oceanic field can lead to an overshoot of MVT and MHT variability, even for relatively weak intensity (0.025 K). This is of importance for prediction systems since it suggests that such an error can push the system beyond its natural attractor. This result is consistent with the analysis of Sévellec and Fedorov (2017). They demonstrated that small spatial-scale disturbances of the density field have important impacts on both MVT and MHT, because MVT and MHT are controlled by local East-West density differences (this result is summarized in the “Appendix”). Consequently, oceanic initial condition uncertainty removes any predictability on short interannual time scales, regardless of its intensity (Fig. 3c1–2). However, these small spatial-scale disturbances disappear quickly because of the relatively fast effect of horizontal diffusion on them (Sévellec and Fedorov 2017). This leads to a “sweet spot” for prediction around 5–10 years for meridional volume and heat transport. This originates from the sharp decrease of the oceanic uncertainty and the relatively slow increase of the atmospheric forcing uncertainty. Hence on interannual to decadal time scales the oceanic initial condition uncertainty dominates the error growth, whereas on decadal to multidecadal time scales the error growth is dominated by the atmospheric stochastic forcing for both MVT and MHT (Fig. 3b1–2). This suggests that MVT and MHT interannual to decadal predictions can be improved by a more accurate oceanic initialization.

It should be noted, however, that in a chaotic turbulent ocean, the sharp decrease in oceanic uncertainty might probably not take place. While the role of a small change in initial condition would still decrease on longer time-scales, stochastic internal noise from ocean turbulence should enhance oceanic uncertainty when time progresses. Nevertheless, this “sweet spot” for prediction that arises in laminar ocean models, suggests that in the real ocean predictions 5–10 years ahead are still probably the most valuables in terms of signal to noise ratio.

## 4 An optimal monitoring system

### 4.1 Method

As mentioned earlier, to decrease the overall uncertainty, and so to increase potential predictability, we can reduce the oceanic initial condition uncertainty (unlike the stochastic atmospheric forcing uncertainty that will remain). This is particularly true for MVT, MHT and OHC, where oceanic initial condition uncertainty dominates the variance growth over interannual time scales. This reduction can be accomplished by a better monitoring system of the ocean state (i.e., accurate measurement of temperature and salinity). Here we show a way to design such an efficient monitoring system.

For this purpose we use the linear optimal perturbation framework. The formulation of LOP is summarized in the "Appendix" and we refer the reader to Sévellec et al. (2007) and Sévellec and Fedorov (2017) for further details. The LOP framework comprises the computation of the pattern of sensitivity to initial conditions for a given linear metric (such as MVT, MHT, SST and OHC). The LOP depends on the lag between the initial condition perturbation and the metric response (examples of LOPs for different lags and for the four cost functions are shown in Figs. 7, 8, 9, 10). As we will demonstrate, the LOPs are directly relevant for the design of monitoring system.

This result has important consequences for the design of efficient monitoring systems. Indeed, increasing measurements in regions of high values of the LOP will decrease the initial condition uncertainty in those regions, hence reducing the re-normalization factor, since the latter is the norm of the LOP to the initial condition uncertainty. This will naturally increase the Predictive Power to its ideal value: \(\text {PP}_\text {Perfect}\) = \(1-\sigma _\text {sto}^2(t)/\sigma _\infty ^2\), since \(\lim _{\delta ^4\rightarrow 0}\sigma _\text {ini}^2(t)\) = 0. This result also demonstrates the usefulness of data-targeting: one should decrease the uncertainty in regions of high intensity of the LOP.

*T*and

*S*are the optimal pattern of the LOP in terms of temperature and salinity from \(\left| {{\varvec{u}}^\mathrm{opt}(t_i)} \right\rangle\) and OOD is the Optimal Observation Density (Fig. 4).

Hence assuming no uncertainty in oceanic initial conditions in regions of high OOD (and modifying \({\mathsf {\varvec{\Sigma }}}_\text {ini}\) accordingly) we see a decrease in error growth, which converges to the perfect case (where oceanic initial condition uncertainty is neglected). This applies to all prediction time scales (Fig. 4a). Significant improvement is especially possible for the MVT with an optimal monitoring system that is quite narrow and mainly located in the Labrador Sea (Fig. 4b1, c1). SST does not show any improvement (Fig. 4a3). This is expected given the strong control of atmospheric stochastic forcing over error growth in SST (Fig. 3b3). MHT and OHC improvement remains possible (Fig. 4a2, a4) but the rather large scale spread of their OOD (Fig. 4b–c2, b–c4) suggests technical difficulty in monitoring accurately such large regions of the ocean.

### 4.2 Applicability to in situ measurements

The knowledge of OOD is extremely useful for fundamental understanding of the sensitivity regions of the ocean. However, it remains the question of the feasibility of its development, even as a guide to future monitoring systems. Hence it is fundamental to relate it to current in situ observational systems and in particular to their technological limitations.

In this context, it is interesting to note the high intensity of OOD below 2000 m (Fig. 4c1–4, especially for OHC metric) which is currently the typical maximum depth of Argo float temperature and salinity measurements. This result, which has already been suggested in a wide range of studies (Wunsch 2010; Dunstone and Smith 2010; Heimbach et al. 2011; Germe et al. 2017b), implies that the maximum improvement in prediction skill can only be gained by also accurately monitoring the deep ocean, as soon possible with the development of Deep Argo floats.

## 5 Discussion and conclusion

In this study, we have focused on the North Atlantic to assess the predictability of four ocean metrics: the AMOC intensity (MVT at 50\(^\circ\)N and above 1500 m depth), the intensity of its heat transport (MHT at 25\(^\circ\)N), the spatially-averaged SST over the North Atlantic (from 30\(^\circ\)N to 70\(^\circ\)N), and the spatial and depth averaged North Atlantic ocean temperature (from 30\(^\circ\)N to 70\(^\circ\)N). Here we propose a theoretical framework to quantitatively assess the growth of small perturbations.

Following the study of Chang et al. (2004), we have developed an exact expression of the ocean predictability for given metrics under 3 main assumptions. (1) The uncertainty remains small (linear assumption); (2) the uncertainty follows a normal distribution (independence of uncertainties); and (3) the ocean dynamics can be treated in a forced context (absence of explicit ocean–atmosphere feedback). In addition, this theoretical result allows us to separate the sources of uncertainty. We are thus able to attribute on a dynamical ground the relative role of internal oceanic initial condition uncertainty and of external atmospheric synoptic noise on ocean prediction uncertainty. After illustrating the method in an idealized model where analytic solutions are known (Hasselmann 1976) the method has been applied to a state-of-the-art GCM (NEMO, Madec et al. 1998) in its 2\(^\circ\) realistic configuration (ORCA2, Madec and Imbard 1996). Given the importance of the model uncertainty on the time scales studied (Hawkins and Sutton 2009), it is worth noting that the single model approach used in this study limits the generalization of our results. Hence our developed framework needs to be applied to other ocean GCMs as well. In particular the location and intensity of deep convection, that is crucial for meridional volume transport (Sévellec and Fedorov 2015) might be strongly model-dependent, potentially modulating the optimal monitoring system.

Our analysis suggests that spatially-averaged SST uncertainty is strongly dominated by the atmospheric synoptic noise (Fig. 3b3), with a strong impact at all time scales, suggesting the limited predictability of this metric. The three other metrics (MVT, MHT and OHC, Fig. 3b1, 2 and 4) are dominated (\(\sim\) 80%) by oceanic initial condition uncertainty on interannual time scales (< 5 years). Whereas the Predictive Power of OHC is monotonically decreasing suggesting the higher predictability of shorter time scales, MVT and MHT predictive power features a “sweet spot” at interannual time scales (Fig. 3c). This means that MVT and MHT are especially predictable on 5–10 years time scales where the signal to noise ratio peaks.

These results were obtained by focusing on the propagation of the error measured through the variance of its probability density function. We have shown that we can also compute solutions for other statistical moments, and so potentially reconstruct the entire probability density function. However, we are limited by two assumptions that restrict the generalization of our solutions. The first assumption is on the structure of the atmospheric stochastic forcing and of oceanic initial condition uncertainty. Assuming a Gaussian white noise leads to simplification in the mathematical treatment of the problem and allows an analytic solution. Hence, by construction our theoretical solution and its numerical application with the ocean GCM consider only random spatially-uncorrelated errors for the oceanic initial condition uncertainties. However initial conditions uncertainties, which are often derived from ocean reanalysis for operational prediction systems, might not have such a useful property. They are often spatially correlated. This might change our results. In particular, this might eliminate the important interannual uncertainties for MHT and MVT due to rapid unbalanced errors. Hence, it would be ideal to generalize our theoretical result to a more general uncertainty. This will be a direction for future investigation. The other assumption is on the linearity of the dynamics which limits the behaviour that can be represented. For instance, the possible occurrence of qualitative changes in the probabilistic distribution (e.g., P-bifurcation), such as the change from an unimodal to a bi-modal distribution, cannot be captured. In this context other methods should be used such as the pullback attractor (e.g., Ghil et al. 2008) or transfer operator (e.g., Tantet et al. 2015) techniques. However they appear to be still computationally expensive and are currently only applicable for idealized models.

The dominance of oceanic initial condition uncertainty in the overall MVT, MHT and OHC uncertainty strongly suggests the possible improvement of predicting these metrics. This source of uncertainty can be reduced by an accurate monitoring of the oceanic state. By using LOPs to design an optimal monitoring system, reduction of such uncertainty is accomplished. Such monitoring systems can be done by the deployment of a suited mooring array or repeated transects at 35\(^\circ\)N and 55\(^\circ\)N. Alternatively, a fleet of gliders might be able to efficiently sample the important regions. We estimate that a fleet of 15 gliders can reduced the uncertainty due to oceanic initial condition by more than 70% on time scales from 1 to 5 years. Also, since gliders are a type of autonomous robotic vehicle, they can both monitor the suggested region and be re-oriented on the fly as the optimal monitoring system evolves. At an operational level our method coupled to a glider fleet would provide a self-adaptative network specifically designed for improving a prediction system.

The spatial resolution of the ocean model used here restricts the study of ocean dynamics to non-eddying regimes. Hence testing the role of oceanic mesoscale eddies is a direction for future work, that would require the extension of the current method to a fully nonlinear framework (following conditional nonlinear optimal peturbations by Mu and Zhang 2006; Li et al. 2014, for instance). The ocean-only forced context of our analysis neglects the impact of large scale atmospheric feedbacks. However, Sévellec and Fedorov (2017) showed that the LOPs are only marginally modified by the ocean surface boundary conditions and Germe et al. (2017a) demonstrated that the overall expected behaviour of LOPs is conserved in a coupled context, despite a reduction of impact. We anticipate that both air-sea interaction and ocean turbulence impact the error growth. The “sweet spot” around 5–10 years for MVT and MHT, showing a minimum in uncertainty in the laminar ocean model, suggests that predicting these time scales in a fully coupled and eddy-resolving climate model is probably most valuable in terms of signal-to-noise ratio and that further development of the method outlined here to more complex Earth System Models is a promising route for improving climate predictions.

## Notes

### Acknowledgements

This research was supported by the Natural and Environmental Research Council UK (SMURPHS, NE/N005767/1 to FS and SSD and DYNAMOC, NE/M005127/1 to AG), FS received support from DECLIC and Meso-Var-Clim projects funded through the French CNRS/INSO/LEFE program, and HAD received support from the Netherlands Center for Earth System Science (NESSC).

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