train that extends from the eastern Paci¬c, across

North America, and into the western Atlantic.

The decorrelation length scale, which is of the

17.4 Teleconnections order of 3000 km, compared well with that of the

observations (see, for example, Thi´ baux [361],

e

17.4.1 Example: 500 hPa Geopotential Height. Fraedrich et al. [127], and Figure 2.8). This length

A classical method for exploring the spatial struc- scale is typical of that of teleconnection patterns

ture of climate variability is to compute cross- that can be computed for other locations in the

correlations between a variable at a ¬xed location midlatitudes of both hemispheres.

and the same or another variable elsewhere. The In contrast, the lower panel of Figure 17.10

resulting map of cross-correlation coef¬cients is displays the teleconnection map that is obtained

called a teleconnection pattern. When the same for a reference point off the coast of Peru at

variable is considered at two nearby locations, the approximately 2—¦ N, 90—¦ W. Here we see that the

correlation will tend to be large and positive (com- entire simulated tropical 500 hPa geopotential

pare with the argument in [1.2.2]). Sometimes height ¬eld varies more or less in unison on the

variables at two well-separated locations are also monthly time scale. Much the same pattern can be

highly”often negatively”correlated.15 obtained for virtually any reference point near the

We demonstrate with DJF monthly mean equator.16 Also note the model™s relatively weak

500 hPa geopotential height from an ensemble rendition of the Paci¬c/North American pattern

of six 10-year GCM simulations. The SST (cf. [13.5.5] and Figure 13.7).

and sea-ice extent were speci¬ed from 1979“ In the following we will present an approach

88 observations so the simulated atmosphere that is used to screen large data sets for such

experienced realistic lower boundary variations teleconnections systematically. It was pioneered

(see [444]).

16 Tropical geopotential height variations are small, and

15 The most prominent example of such a teleconnection is primarily re¬‚ect variations in the temperature of the lower

the Southern Oscillation discussed in [1.2.2]. tropical troposphere.

17.4: Teleconnections 383

by Wallace and Gutzler [409] and Horel and Wal- negatively weighted:

lace [182]. See also the review by Navarra [290], 1

PNA = z 20—¦ N ,160—¦ W ’ z 45—¦ N ,165—¦ W

the example discussed in [2.8.8], and Figure 2.8.

4

+ z 55—¦ N ,115—¦ W ’ z 30—¦ N ,85—¦ W

17.4.2 The Wallace and Gutzler Approach. (17.36)

Let X represent a gridded data variable, such

as SLP or 500 hPa geopotential height, and let where z is 500 hPa geopotential height. Similar in-

Rx x be the corresponding matrix of estimated dices are obtained for the other four teleconnection

cross-correlations. The jth column of Rx x , ρ j , patterns.

Such indices may be used to derive composites

contains the estimated cross-correlations between

X j and X. Thus, if X is m-dimensional, Rx x [17.3.2] or associated correlation patterns [17.3.4],

can be shown as m maps. All maps have unit to monitor the strength of a teleconnection, or

value at the base point j, and for most variables they can be fed into predictive schemes. Also

they will have relatively large positive values in a the correlation between teleconnections can be

neighbourhood of the base point. Points that are quanti¬ed. Wallace and Gutzler found moderate

outside the ˜region of in¬‚uence™ of the base point correlations between patterns with spatial overlap

are not considered interesting. Such correlation and small correlations between patterns with little

or no spatial overlap.

maps are called teleconnection patterns.

The teleconnections patterns depend somewhat

In the next subsection we will present some

results from Wallace and Gutzler™s [409] original on the choice of the base point. The base point

analysis, and then de¬ne a measure of the strength for the PNA pattern shown in the upper panel of

Figure 3.9 was (45—¦ N, 165—¦ W). Similar patterns

of the teleconnections in [17.4.4].

are obtained if another centre of action, (20—¦ N,

160—¦ W), (55—¦ N, 115—¦ W), or (30—¦ N, 85—¦ W), is

17.4.3 The PNA, WA, WP, EA, and EU-Family used as the base point.

of Teleconnections. We brie¬‚y discussed Wal-

lace and Gutzler™s [409] identi¬cation of charac-

teristic patterns of the month-to-month variability

of winter 500 hPa height in [3.1.6]. Five ˜sig-

ni¬cant™17 correlation maps with sequences of

large positive and negative centres of action were

found. These patterns, called the Eastern Atlantic

(EA), Paci¬c/North American (PNA), Eurasian

(EU), West Paci¬c (WP), and West Atlantic (WA)

patterns, are shown in Figure 3.9.

The locations of maxima and minima of

a teleconnection pattern (i.e., the centres of

action) can be used to de¬ne time-dependent

teleconnection indices. Wallace and Gutzler de¬ne

such an index as a weighted sum of the heights at

the centres of action. In the PNA case, the centres

of action are located at (20—¦ N, 160—¦ W), (55—¦ N,

115—¦ W), (45—¦ N, 165—¦ W) and (30—¦ N, 85—¦ W). The

¬rst two points are associated with maxima and the

last two with minima. Thus the contribution from

Figure 17.11: Teleconnectivity map of 500 hPa

the last two centres of action in the PNA index is

height in northern winter. From [409].

17 Wallace and Gutzler use the word ˜signi¬cant™ in a

somewhat pragmatic sense. They split the data set into two

subsets, and used one subset to establish the teleconnection

17.4.4 The ˜Teleconnectivity.™ A typical feature

patterns and the second subset to assess (successfully) the

of teleconnection maps is the presence of large

stability of the patterns. In this way they determined a rule-

of-thumb that correlations |ρ| > 0.75 should be reproducible negative correlations. It therefore makes sense to

in samples of size 15. Reproducibility is a stronger criterion

de¬ne the ˜teleconnectivity™ T j of a base point j

that statistical signi¬cance since |ρ| > 0.5 is suf¬cient to

as the maximum of all negative correlations:

reject H0 : ρ = 0 at approximately the 5% signi¬cance level

(see [8.2.3] and David [100]). Wallace and Gutzler also found

T j = ’ min ρ i j . (17.37)

consistent patterns with an EOF analysis (see Chapter 13). j

17: Speci¬c Statistical Concepts

384

T j can then be plotted as a spatial distribution;

Wallace and Gutzler™s example is shown as

6

Figure 17.11. We occasionally ¬nd that the

4

maxima in the teleconnectivity map are connected

2

by a common point, that is, that T j and Tk obtain

0

their values from ρ l j and ρ lk for a common

-2

-4

point l. These conditions, which hint at physical

relationships, can be displayed with arrows as in 0 50 100 150 200

Figure 17.11.

Figure 17.12: A realization of an AR(1) time series

17.4.5 Generalizations. The basic idea of with lag-1 correlation coef¬cient ±1 = 0.9 (solid

mapping correlations between one variable at a curve), and its 11-term running mean (dashed

base point and another variable at many other curve).

geographically distributed points can be applied

to any two climate variables. Indeed the square

Another way to assess reproducibility is to test

correlation matrix R can be replaced with a

rectangular cross-correlation matrix Rzx . Also, H0 : ρ = 0 at every point in a teleconnection