#
`nao` documentation

`nao` calculates the North Atlantic Oscillation index from sea-level pressures based on the definition proposed by Hurrell, 1995. For all practical purposes the NAO index is equivalent to the Arctic Oscillation as well as the North Annular Mode. For more information see this introduction to Annular Modes by David Thompson at NCAR.

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## Contents

## Syntax

idx = nao(slpA,slpB,t)

## Description

`idx = nao(slp40,slp65,t)` calculates the North Atlantic Oscillation index from two time series of sea-level pressures at two stations (A and B) and their corresponding times `t`. Here, Station A (either Azores or Lisbon or Gibraltar) is usually south of Station B (Iceland).

## Example

Here, we'll recreate Figure 4 from Jones et al.'s classic 1997 paper, Extension to the North Atlantic Oscillation using early instrumental pressure observations from Gibraltar and South-west Iceland (Jones et al., 1997), which depicts the monthly NAO index. This dataset builds on Hurrell's 1995 paper and station data can be found https://crudata.uea.ac.uk/cru/data/nao/index.htm. The dataset we utilize here goes from January, 1865 to December, 2017.

## Load Data

We can load data containing zonal-mean SLP at 40S and 65S calculated from observations from 1957 January to 2018 December.

```
load nao_slp_data.mat
```

## Plot pressure data

The `nao_slp_data.mat` file contains monthly mean SLP data from 1865 to 2018 for Gibraltar (A) and Iceland (B). See above to find where you can obtain this dataset. Here is the pressure data we'll use to calculate the NAO:

figure plot(t,slpA); hold on plot(t,slpB) axis tight legend('SLP at Gibraltar','SLP at SW Iceland')

## Calculate NAO index

The `nao` function normalizes each time series relative to the full baseline dataset and differences the two pressure anomaly time series to yield the NAO index. We'll calculate NAO, then smooth it with a 12 month moving average (see the note below on filtering):

% Calculate NAO index: nao_idx = nao(slpA,slpB,t); % Apply a moving-mean filter: nao_idx_f = movmean(nao_idx,12); figure plot(t,nao_idx_f,'k','linewidth',1) ylim([-3 3]) % sets vertical limits hline(0,'k-') % draws horizontal line set(gca,'xaxislocation','top') % to match Jones 1997

Above, the built-in Matlab function `movmean` was used to calculate the 12 month moving mean, and a horizontal dashed line was drawn with `hline` to match Marshall's Figure 7.

## A note on filtering

The caption to Figure 4 in Jones et al., 1997 indicates that a 12-month Gaussian filter was used to smooth the time series before plotting. A Gaussian filter is a type of moving-average filter, and it's weighted such that values near the edges of the moving window don't contribute much to the average. The weighting of a Gaussian filter window takes the shape of a Gaussian curve, in contrast to an ordinary moving-average filter whose weighting window is effectively a rectangle wherein all values contribute equally to the average.

Unfortunately, the Jones et al. description of a "12-month Gaussian filter" is somewhat ambiguous, because it's unclear whether that means the total width of the Gaussian window, or the 1-sigma width, or twice the 1-sigma width, or something else entirely. How exactly do you define the width of something that should trail that takes an infinite distance to gradually trail off to zero? Sometimes people talk about Gaussian filters in terms of 2*pi*sigma, because that's the distance at which the weighting factor is equal to exp(-0.5), but it's unclear if that's what Jones et al. intended.

We don't really have a clear answer on the exact width of the Gaussian window, or its exact shape; however, in this particular case, the details of the filter design probably do not matter very much. We used a plain moving-average above, and still got a plot that looks pretty much the same as Figure 4 of Jones et al. That tells us that while it would be nice if we could replicate the methods of Jones et al. exactly, their methods and explanations were enough to replicate their results generally.

## References

Hurrell, J.W., 1995: Decadal Trends in the North Atlantic Oscillation: Regional Temperatures and Precipitation. *Science* Vol. 269, pp.676-679 doi:10.1126/science.269.5224.676

Jones, P. D. et al., 1997: Extension to the North Atlantic oscillation using early instrumental pressure observations from Gibraltar and south-west Iceland. Int. *J. Climatol.*, 17: 1433-1450. https://doi.org/10.1002/(SICI)1097-0088(19971115)17:13%3C1433::AID-JOC203%3E3.0.CO;2-P

## Author Info

The `nao` function and supporting documentation were written by Kaustubh Thirumalai for the Climate Data Toolbox for Matlab, 2019.