The frequency with maximum coherency approxi- wide range of time scales. Their maxima coincide

mates the oscillation frequency ·. These frequen- well with · in both cases, although agreement

cies coincide exactly only when u is zero. In is slightly better for the process with the lower

general, they are different because 2r/(1+r 2 ) < 1. damping rate (r = 0.8).

11.5: Frequency“Wavenumber Analysis 241

Figure 11.10: Power spectra x x = yy (heavy line, left hand axis) and coherency spectra κx y (light

line, right hand axis) of four bivariate AR(1) processes with parameter matrices A given by (11.80). See

text for details.

11.4.9 The POP Process: The Role of v. The opposite interpretation holds when v < 0:

Parameter v in the rotation matrix A (11.80) X lags Y and the characteristic sequences are

determines the direction of rotation. Suppose, for 0 1 0

... ’ ’ ’

convenience, that there is no damping. When ’1

1 0

v > 0, repeated application of A smoothly (11.84)

’1 0 1

transforms the initial state (x, y) = (1, 0) into ’ ’ ’ ’ ...

0 1 0

the state (x, y) = (0, 1) in a quarter of a period

T . Continued application of A then transforms These ideas resurface in the next section when we

(x, y) = (0, 1) into (x, y) = (’1, 0) in the deal with eastward and westward travelling waves.

next quarter period. Thus, for positive numbers

v, the system tends to create sequences in the

11.5 Frequency“Wavenumber

(X,Y)-space of the form

Analysis

11.5.1 Introduction. Wave-like processes play

an important role in the dynamics of geophysical

’1 ¬‚uids. Physical processes exhibit standing waves

1 0

... ’ ’ ’ with maxima, minima, and nodes at ¬xed

0 1 0

(11.83) locations, propagating waves with wave crests

0 1 0

’ ’ ’ ’ ... that move in space, and mixed forms that are a

’1 0 1 combination of the two.

Waves are often readily described with trigono-

metric functions such that at any given time t the

wave ¬eld f (x, t) can be expanded into sines and

The sign of v does not affect the auto-

cosines as

and coherency spectra shown in Figure 11.10. ∞

2π kx

However, the phase spectrum is affected. When v f (x, t) = ckt cos (11.85)

L

is positive the phase spectrum (11.81) is positive k=0

(for positive frequencies ω), which is consistent 2π kx

+ skt sin

with the interpretation that X leads Y. L

11: Parameters of Univariate and Bivariate Time Series

242

3 The cross-spectrum is separated into compo-

where L is some reference length such as the

nents representing the variance of eastward

circumference of Earth at a given latitude. The

and westward travelling waves.

coef¬cients ckt and skt are given by

L

f (x, t) d x k=0 The methods used to perform the separa-

0

ckt = tion are derived using heuristic arguments.20

L

f (x, t) cos( 2πLkx ) d x k > 0

2 The total variance is assumed to consist

0

k=0 of only ˜eastward™ and ˜westward travel-

0

skt = L

f (x, t) sin( 2πLkx ) d x k > 0. ling™ variance. The total variance is split

2 0

up into equal contributions from eastward

Index k is known as the wavenumber. The time- and westward ˜travelling waves™ when the

dependent coef¬cients ckt and skt sometimes processes are generated by white noise, or

oscillate with a period that is conditional upon by non-propagating features (see examples

the wavenumber k (e.g., Rossby waves). Functions in [11.5.5]). This seems reasonable when

that relate the variation of the period with there are standing features that can be thought

the wavenumber are commonly referred to as of as the sum of coherent waves that prop-

dispersion relations because they relate a spatial agate in opposite directions. However, one

scale, namely L/k, to a time scale. might be skeptical about applying this ap-

It is useful to look for dispersion relationships proach to stochastic processes since white

in observed data, either to support a dynamical noise, for example, does not contain ˜travel-

theory that predicts dispersion relationships or as a ling waves.™ We can live with these ambigui-

diagnostic that may ultimately lead to the detection ties in the scienti¬c lexicon if the limitations

of wave-like dynamics. are asserted and understood. However, use of

Frequency“wavenumber analysis, or space“ this slang without also presenting the caveats

time spectral analysis, is a tool that can be used leaves plenty of opportunity to misinterpret

to diagnose possible relationships between spatial results.

and time scales. The original concept, developed

by Deland [102] and Kao [211, 212], assumed

4 Additional heuristic arguments are used to

that the wave ¬eld evolved in a deterministic way.

assign a part of the overall variance to

Hayashi [169, 170] and Pratt [319] adapted the

standing waves.

method by accounting for the stochastic nature of

Pratt [319] interprets the modulus of the

the analysed ¬elds.

difference between westward and eastward

There are many examples of applications of

travelling wave variance as ˜propagating var-

the frequency“wavenumber analysis. For example,

iance™ and labels the remainder as ˜standing

Hayashi and Golder [171] studied the Madden-

wave™ variance. With this interpretation, the

and-Julian Oscillation with this tool. Also, many

standing variance comprises all truly standing

workers, including Fraedrich and coworkers [124,

waves plus all random ¬‚uctuations. Depend-

126] and Speth, Madden, and others [353, 354,

ing upon the sign of the difference between

419], have analysed the frequency-wavenumber

the eastward and westward travelling wave

spectrum of the extratropical height ¬eld.

variance, the propagating variance is inter-

preted as being either purely ˜eastward™ or

11.5.2 The Four Steps. Frequency-wavenum-

˜westward™ variance.

ber analysis is performed in four steps.

Hayashi [170] attributes the coherent part

1 The ¬eld of interest is expanded into a series

of the eastward and westward travelling

of sine and cosine functions (11.85).

variance to standing waves. The incoherent

The ¬eld (e.g., an atmospheric process on part is interpreted as eastward or westward

a latitude circle), is assumed to be spatially propagating variance. Thus the total variance

periodic. is split up into three compartments: stand-

ing waves, eastward propagating waves, and

2 The bivariate time series, composed of

westward propagating waves. The propagat-

the time-dependent sine and cosine coef¬-

ing variance is described by a two-sided

cients ckt and skt , is assumed to be a random

realization of a bivariate stochastic process.

20 The adjective ˜heuristic™ describes an argument that is not

The cross-spectrum of this process is esti- rigorously logical or complete and may be supported by ad-hoc

mated. assumptions.

11.5: Frequency“Wavenumber Analysis 243

Using (11.19), we see that the total variance21 at

spectrum and the standing variance by a

one-sided spectrum. wavenumber k can be re-expressed as

Var(Ct ) + Var(St )