# The North Atlantic Oscillations: Lead–Lag Relations for the NAO, the AMO, and the AMOC—A High-Resolution Lead–lag Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials

^{15}N [6]. The data were obtained from Levke Caesar, Potsdam Institute for Climate Impact Research. The AMOC is the variable that most directly expresses transfer of heat (down to 2000 m, National Oceanography Centre, Available online: https://www.rapid.ac.uk/rapidmoc/ (accessed on 15 February 2022). The AMOC has been instrumentally observed at 26° N since 2004, the Rapid Climate Change Programme, RAPID. We compare two versions of the AMOC time series in Appendix A and we examine our choice of AMOC series in the discussion section. The AMOC series have unit Sv.

## 3. Methods

_{1}and v

_{2}through three consecutive observations is calculated from Equation (1):

_{1}× v

_{2})⋅”Arccos” ((v

_{1}⋅v

_{2})/|v

_{1}||v

_{2}|)

_{i}− y

_{i−1})/(x

_{i}− x

_{i−1}) with i = 2, 3, …

## 4. Results

#### LL Relations, Cycle Periods, and Phase Shifts

## 5. Discussion

#### 5.1. Cycle Periods

#### 5.2. The AMOC–AMO Relations

#### 5.3. The AMOC–NAO Relations

#### 5.4. The AMO–NAO Relations

#### 5.5. The Ekman and the AMOC/NAO Relations

#### 5.6. The Bidecadal Periods

#### 5.7. Robustness

#### 5.8. Future Work

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. AMOC Series

**Figure A1.**AMOC time series. The regression lines (straight lines) show regressions from 2009 to 2019; that is, the last decade. The Caesar, McCarthy [6] time series (black line) are annual averages. The blue line shows cross-year average values from March one year to April next year, and with the Ekman (wind-driven) component removed. The Acis (UK) values are read from a graph. The Caesar data were kindly supplied by Levke Caesar, Maynooth University.

**Figure A2.**LL relations between AMOC and NAO (DJF). (

**a**) Time series for AMOC and NAO (all months) and only the winter months (D, J, F); (

**b**) LL relations. Note that the black bars are divided with 10 and are thus almost all significant. The period where AMOC lags NAO is from 1961 to 1988.

## Appendix B. Example

**Figure A3.**Example: AMOC LOESS(0.3)-smoothed (x) and shifted forward six years (y). (

**a**) AMOC time series, original (blue) and shifted 6 years forward. The original annual AMOC series has been LOESS(0.3)-smoothed. The zigzag curve indicates the length of years found by the cumulative angle method. (

**b**) LL relations between AMOC original and AMOC shifted 6 years forward, LL(AMOC, AMOC+6). The black bars show Ang(3); that is, LL relation over three consecutive observations in the paired time series. The grey bars show LL(9); that is, the relation between positive and negative angles over 9 consecutive observations. (

**c**) Phase plot for the pairs AMOC and AMOC+6. Note that most rotations are counterclockwise (positive, +) showing that AMOC leads AMOC +6. (

**d**) Phase shifts calculated relative to moving cycle period and with average cycle period. Average cycle period is about 6 years, corresponding to the six years design phase shift. Lowe curve shows moving β-coefficient.

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**Figure 1.**Method explanation. (

**a**) Part of sine functions y = sin (0.1 (t + δ)) where δ is −5 (red curve) and +5 (blue curve). The black, bold curve is the target cure that we want to examine LL relations for t = 0 to 25. (

**b**) Phase plot for pair of curves. Red curve shows the leading sine function, and the blue curve shows the lagging sine function. (

**c**) The LL relation and the correlations between cyclic time series with common cycle period, λ, as a function between the phase shifts, δ, between them.

**Figure 2.**AMOC, AMO and NAO. (

**a**) Time series raw and LOESS(0.3)-smoothed. The detrended and LOESS(0.3)-smoothed versions of AMOC shifts sign from starting with (+) in 1947, then 1969, 1997, 2010. AMO starting from (+) in 1947, 1996, 1999. NAO starting from (+) in 1947, 1952, 1972, 1997, 2014, 2020. (

**b**) Power spectral density. Peaks at 3, 10, 17, 32, 47, 57, and 62 years.

**Figure 3.**LL relations for AMOC and AMO. (

**a**) Time series raw. The upper zigzag curve shows cycle periods. (

**b**) LL relations between AMOC and AMO. Black bars show angles, θ(3), calculated over three consecutive observations; grey bars show the sign of angles, LL(9) calculated over 9 years. Dashed lines show confidence interval. Droplines designate time windows of 20 years starting at 1960. (

**c**) Phase shift calculated with two methods and the β-coefficient for the two series. (

**d**) Phase plot for the two series with AMOC and AMO slightly LOESS(0.1)-smoothed on the x- and y-axis, respectively. (

**e**) The two series LOESS(0.3)-smoothed and their cycle periods. (

**f**) LL relations between the two series. Black bars for θ(3), and grey bars for LL(9).

**Figure 4.**LL relations for AMOC and NAO. (

**a**) Time series raw. The upper zigzag curve shows cycle periods. (

**b**) LL relations between AMOC and NAO. Black bars show angles, θ(3), calculated over three consecutive observations; grey bars show the sign of angles, LL(9) calculated over 9 years. Dashed lines show confidence interval. Droplines designate time windows of 20 years starting at 1960. (

**c**) Phase shift calculated with two methods (Thin line: average cycle period; bold line: running average cycle period) and the β-coefficient for the two series. (

**d**) Phase plot for the two series with AMOC and NAO slightly LOESS(0.1)-smoothed on the x- and y-axis, respectively. (

**e**) The two series LOESS(0.3)-smoothed and their cycle periods. (

**f**) LL relations between the two series. Black bars for θ(3), and grey bars for LL(9).

**Figure 5.**LL relations for AMO and NAO. (a) Time series raw. The upper zigzag curve shows cycle periods. (b) LL relations between AMO and NAO. Black bars show angles, θ(3), calculated over three consecutive observations; grey bars show the sign of angles, LL(9) calculated over 9 years. Dashed lines show confidence interval. Droplines designate time windows of 20 years starting at 1960. (c) Phase shift calculated with two methods (Thin line: average cycle period; bold line: running average cycle period) and the β-coefficient for the two series. (d) Phase plot for the two series with AMO and NAO slightly LOESS(0.1)-smoothed on the x- and y-axis, respectively. (e) The two series LOESS(0.3)-smoothed and their cycle periods. (f) LL relations between the two series. Black bars for θ(3), and grey bars for LL(9).

**Figure 6.**Summary characteristics. (

**a**) Principal component loading plot for LL relations between the three modes, AMOC, AMO, and NAO, their high- and low-frequency series. (

**b**) PCA loading plot for the series β-coefficients (slopes between paired series), their high- and low-frequency components. (

**c**) The stacked sum of the absolute value of the LL (x,y) values. The droplines designate troughs in 1960–1961, 1968–1969, 1977, 1994, and 1999–2001. (

**d**) Ekman time series, raw series (dashed lines) and Ekman 26

^{,}LOESS (0.2)-smoothed, and NAO LOESS(0.1)-smoothed (Full lines). The red bars show LL relations based on the raw data, and the light blue bars show LL relations based on LOESS-smoothed series.

**Figure 7.**LL relations between the AMOC, the AMO, and the NAO: LL(AMOC, AMO), LL(AMOC, NAO), LL(AMO, NAO). AMOC is measured in Sverdrup, Sv; AMO is measured in temperature degrees; and NAO is measured in sea-level pressure, hPa. We have distinguished three periods corresponding to the LL patterns in Figure 2. The arrows indicate leading directions, and arrow thickness indicates leading strength, Equation (2). The numbers in each corner show numerical values for direction (LL(x, y) > 0: x→y. The leading strength is the average for the ten mid-years in each period. Upper row: high-frequency oscillations. Lower row: low-frequency oscillations. Red arrows suggest that the leading role is persistent across all time periods.

**Table 1.**Characteristics of Lead–lag relations between pairs of North Atlantic climate modes. The bold numbers in the two last columns suggest that for the corresponding pairs, identifying LL relations with cross-correlation techniques may give results that are similar to ours.

Series 1 | Series 2 | Smoothing Parameter, p | Cycle Period Years | Phase Shift | LL(1,2) | β-Coefficient | |
---|---|---|---|---|---|---|---|

average | max | years | % counter clockwise | % positive | |||

AMOC | AMO | 0 | 5.6 | 13 | 1.4–1.5 | 36 | 68 |

0.3 | 17 | 26 | 1.6–2.1 | 56 | 82 | ||

AMOC | NAO | 0 | 2.9 | 8 | 0.78–1.07 | 56 | 34 |

0.3 | 23 | 43 | 7.6–9.5 | 25 | 7 | ||

AMO | NAO | 0 | 2.3 | 7 | 0.77–1.3 | 50 | 5 |

0.3 | 2.3 | 44 | 6.5–7.1 | 33 | 32 | ||

Ekman | NAO | 0.2/0.1 | 4 | 5 | 0.7–0.8 | 60 | 80 |

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**MDPI and ACS Style**

Seip, K.L.; Wang, H.
The North Atlantic Oscillations: Lead–Lag Relations for the NAO, the AMO, and the AMOC—A High-Resolution Lead–lag Analysis. *Climate* **2022**, *10*, 63.
https://doi.org/10.3390/cli10050063

**AMA Style**

Seip KL, Wang H.
The North Atlantic Oscillations: Lead–Lag Relations for the NAO, the AMO, and the AMOC—A High-Resolution Lead–lag Analysis. *Climate*. 2022; 10(5):63.
https://doi.org/10.3390/cli10050063

**Chicago/Turabian Style**

Seip, Knut Lehre, and Hui Wang.
2022. "The North Atlantic Oscillations: Lead–Lag Relations for the NAO, the AMO, and the AMOC—A High-Resolution Lead–lag Analysis" *Climate* 10, no. 5: 63.
https://doi.org/10.3390/cli10050063