Decadal predictability of the Atlantic meridional overturning circulation and climate in the IPSL-CM5A-LR model

Abstract

This study explores the decadal potential predictability of the Atlantic Meridional Overturning Circulation (AMOC) as represented in the IPSL-CM5A-LR model, along with the predictability of associated oceanic and atmospheric fields. Using a 1000-year control run, we analyze the prognostic potential predictability (PPP) of the AMOC through ensembles of simulations with perturbed initial conditions. Based on a measure of the ensemble spread, the modelled AMOC has an average predictive skill of 8 years, with some degree of dependence on the AMOC initial state. Diagnostic potential predictability of surface temperature and precipitation is also identified in the control run and compared to the PPP. Both approaches clearly bring out the same regions exhibiting the highest predictive skill. Generally, surface temperature has the highest skill up to 2 decades in the far North Atlantic ocean. There are also weak signals over a few oceanic areas in the tropics and subtropics. Predictability over land is restricted to the coastal areas bordering oceanic predictable regions. Potential predictability at interannual and longer timescales is largely absent for precipitation in spite of weak signals identified mainly in the Nordic Seas. Regions of weak signals show some dependence on AMOC initial state. All the identified regions are closely linked to decadal AMOC fluctuations suggesting that the potential predictability of climate arises from the mechanisms controlling these fluctuations. Evidence for dependence on AMOC initial state also suggests that studying skills from case studies may prove more useful to understand predictability mechanisms than computing average skill from numerous start dates.

Appendices

Appendix 1: diagnostic potential predictability (DPP) approach

The DPP approach attempts to quantify the fraction of long-term variability (considered as predictable) as compared to the internal variability (considered as chaotic and unpredictable). The long-term variability that rises above this noise is deemed to arise from processes operating in the physical system that are assumed to be, at least potentially, predictable. Boer (2004) defined the potential predictability variance fraction (ppvf) as an estimate of DPP. Here, we use its non-biased estimation (see Boer 2004 for further details) defined as:

$$ ppvf = \frac{{\sigma_{N}^{2} - \frac{1}{N}\sigma^{2} }}{{\sigma^{2} }} $$

(1)

where σ ² _N represents the variance of N-year means, and σ² represents the interannual variance of the variable X considered. The ppvf varies between 0 and 1; a ppvf close to 0 implies no long-term variability and thus no potential predictability. Conversely, ppvf close to 1 implies large predictability. Statistical significance of ppvf is judged using a F-test at the 95 % confidence level. A threshold for “useful” potential predictability is however hard to define, as it is likely to be purpose and situation dependent. Nevertheless, it remains an easy statistical way to estimate the average predictive skill in a model.

Prognostic potential predictability (PPP) approach

Prognostic predictability studies consist in performing ensemble experiments with a single coupled model by perturbing the initial conditions (ICs) supposed to represent atmospheric chaotic noise or uncertainty in the present climate state. Ensemble Correlation (EC) and Ensemble spread (ES) are the two deterministic measures used here to quantify the predictability of the simulated climate.

2.1 Ensemble correlation (EC)

In the forecast framework, correlation addresses the question: “to what extent are the forecasts varying coherently with the observed variability?”. In the “perfect ensemble” approach, the definition of “observed variability” differs from one study to another: it can refer to the variability of the ensemble mean (i.e. the average of all members for an individual ensemble) as in Msadek et al. 2010 (M10) or to the variability of an individual member as in Collins and Sinha 2003 (CS03). In other words, in the M10 approach, predictability skill is evaluated by correlating each member of the ensemble to the ensemble mean whereas in the CS03 approach each member is correlated to each other. If M is the number of members, we therefore obtain M (resp. M(M-1)/2) individual correlations for M10 (resp. CS03). Independently of the approach used, the formula for the individual correlation of any pairs p is:

$$ r_{p} = \frac{{\left[ {T\sum\nolimits_{t = 1}^{t = T} {A_{t} B_{t} } } \right] - \left[ {\sum\nolimits_{t = 1}^{t = T} {A_{t} \sum\nolimits_{t = 1}^{t = T} B_{t} } } \right]}}{{\sqrt {\left[ {T\sum\nolimits_{t = 1}^{t = T} {A_{t}^{2} - \left( {\sum\nolimits_{t = 1}^{t = T} {A_{t}^{2} } } \right)} } \right]} \left[ {T\sum\nolimits_{t = 1}^{t = T} {B_{i}^{2} - \left( {\sum\nolimits_{t = 1}^{t = T} {B_{t} } } \right)^{2} } } \right]}} $$

(2)

where T is the number of years over which we want the correlation for, and A and B are the members forming the pair p. Once the individual correlations of all pairs have been calculated (M pairs for M10, M(M-1)/2 pairs for CS03), EC of the ensemble is computed as the mean of all individual correlations through a Fisher Transformation (Fisher 1921). We will consider the two definitions presented above to evaluate possible differences in their respective score of predictive skills. Statistical significance of the resulting EC is judged using a one-tailed Student’s t-distribution test at the 90 % confidence level with degree of freedom corresponding to the average degree of freedom of all individual correlations. The degree of freedom of these latter takes into account the persistence in the two timeseries following Bretherton et al. (1999). In order to gain confidence in the estimation of the EC significativity, we also evaluated its significance by using the “field significance” approach (e.g. Livezey and Chen 1983). The statistical significance of EC obtained with this test is very similar to the ones obtained from the average degree of freedoom of all individual correlations.

2.2 Ensemble spread (ES)

ES or Root Mean Squared Error RMSE or again the Mean Squared Skill Score MSSS (as defined by the US CLIVAR working group on Decadal Predictability, http://clivar-dpwg.iri.columbia.edu) addresses the question: “how large are the typical errors in the forecast (among members) relative to those implied by baseline?”. Consistently with EC, we consider the two definitions of the baseline which arise from the literature: for a given lead-time LT, ES of an ensemble of individual members i is defined respectively as:

$$ ES_{M10} (LT) = \sqrt {\frac{1}{M}\sum\limits_{i = 1}^{M} {(X_{i} (LT) - \bar{X}(LT))^{2} } } $$

(3)

$$ ES_{CS03} (LT) = \sqrt {\frac{2}{M(M - 1)}\sum\limits_{i = 1}^{M} {\sum\limits_{j = i + 1}^{M} {(X_{i} (LT) - X_{j} (LT))^{2} } } } $$

(4)

where we define: $\bar{X}(LT) = \frac{1}{M}\sum\nolimits_{i = 1}^{M} {X_{i} (LT)}$

We demonstrate below that there actually exists a relationship of proportionality between (3) and (4). Let consider the two following definitions of Mean Squared Error:

$$ E_{M10} = \frac{1}{M}\sum\limits_{i = 1}^{M} {(X_{i} - \bar{X})^{2} } $$

(5)

$$ E_{CS03} = \frac{2}{M(M - 1)}\sum\limits_{i = 1}^{M} {\sum\limits_{j = i + 1}^{M} {(X_{i} - X_{j} )^{2} } } $$

(6)

By expanding $(X_{i} - \bar{X})^{2}$ in (5) and after a few rearrangements we show that:

$$ E_{M10} = \overline{{X^{2} }} - \bar{X}^{2} $$

(7)

Then, if we introduce:

$$ E = \frac{2}{M(M - 1)}\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {(X_{i} - X_{j} )^{2} } } $$

(8)

We show by a recurrence reasoning that:

$$ E = 2E_{CS03} $$

(9)

By expanding (X _i  − X _j )² in (8) and after a few rearrangements, we show that:

$$ E = \frac{4}{M - 1}(\overline{{X^{2} }} - \bar{X}^{2} ) $$

(10)

By combining (7), (9) and (10), we obtain the following relationship:

$$ E_{CS03} = \frac{2M}{M - 1}E_{M10} $$

(11)

Therefore

$$ ES_{CS03} (LT) = \sqrt {\frac{2M}{M - 1}} ES_{M10} (LT) $$

(12)

And there exists a factor of proportionality $\sqrt {\frac{2M}{M - 1}}$ between the ensemble spread of both CS03 and M10 definitions.

Generally, the trajectories of individual members diverge with time and thus ES increases with LT. When ES saturates at the control RMSE, we consider that there is no more potential predictability: the spread of the forecast is of similar magnitude as the natural spread of the modelled climate, and no predictability can be inferred. In CS03 (M10) the control RMSE is defined as σ√2 (σ√[(M − 1)/M]), where σ is the standard deviation of the control integration. Statistical significance of ES as compared to the respective threshold (or control RMSE) is judged using a F-test at the 95 % confidence level. The maximum LT at which a variable is said to be potentially predictable is the last LT before ES persistently exceeds the threshold.

2.3 Relationship between ES and EC

We consider here centred and normalized (by the standard deviation) data in time t.

We consider the CS03 definition of ES and EC:

$$ ES^{2} = \frac{2}{M(M - 1)}\sum\limits_{i = 1}^{M} {\sum\limits_{j = i + 1}^{M} {(X_{i} (t) - X_{j} (t))^{2} } } $$

(13)

$$ EC = \frac{2}{M(M - 1)}\sum\limits_{i = 1}^{M} {\left( {\sum\limits_{j = i + 1}^{M} {corr(X_{i} (t),X_{j} (t))} } \right)} $$

(14)

where the discrete time correlation using centred and normalized data is:

$$ {\text{corr}}(X_{i} (t),X_{j} (t)) = \frac{1}{T}\sum\limits_{t = 1}^{T} {X_{i} } (t)X_{j} (t) $$

We consider the average of ES over the period of time T: $< ES^{2} >_{T} = \frac{1}{T}\sum\nolimits_{t = 1}^{T} {ES^{2} } (t)$

By expanding (X _i (t) − X _j (t))² in (13) and after a few rearrangments we can show that:

$$ < ES^{2} >_{T} = \frac{1}{T}\sum\limits_{t = 1}^{T} {\frac{2}{M(M - 1)}} \sum\limits_{i = 1}^{M} {\sum\limits_{j = i + 1}^{M} {(X_{i}^{2} (t) + X_{j}^{2} (t)} } - 2X_{i} (t)X_{j} (t)) $$

$$ < ES^{2} >_{T} = \frac{2}{M(M - 1)}\sum\limits_{i = 1}^{M} {\sum\limits_{j = i + 1}^{M} \frac{1}{T} \sum\limits_{t = 1}^{T} {\left( {X_{i}^{2} (t) + X_{j}^{2} (t)} \right)} } - \frac{4}{M(M - 1)}\sum\limits_{i = 1}^{M} {\sum\limits_{j = i + 1}^{M} \frac{1}{T} \sum\limits_{t = 1}^{T} {X_{i} (t)X_{j} (t)} } $$

Since the variables are centred and normalized:

$$ \frac{1}{T}\sum\limits_{t = 1}^{T} {X_{i}^{2} (t) + X_{j}^{2} (t)} = 2 $$

Hence, we obtain the following result:

$$ < ES^{2} >_{T} = 2(1 - EC) $$

For the real case, where the data are not normalized and centred, which is more appropriate for ES estimation, no such simple relationship can be found analytically, but we hypothesize that ES and EC remain related. A few illustrations of such link are evidenced in the manuscript (see Sect. 4.1.2) and plead in favour of this hypothesis.

Appendix 2

We present the individual correlation maps for temperature (Fig. 11) and precipitation (Fig. 12) for each starting date in the PPP protocol. These individual results are aggregated in Figs. 9 and 10.

Fig. 11

Surface temperature. Colours represent EC computed as in CS03 for each starting date and years 1–10 (left panels), 1–20 (right panels) of each ensemble experiment. Areas where the correlation is not statistically significant at the 90 % level are shown in white. Dots represent grid points where the ES is statistically significantly smaller than the control RMSE at the 95 % level

Fig. 12

As Fig. 11 for precipitation