the wave crest will be replaced by a trough.

When the process is stochastic, we can no longer 1

fw

2

σE = cs (ω) dω

2

assume that the eastward movement is strictly

0

uniform or that a well-de¬ned period exists.

0

However, if the waves tend to travel eastward, fw

σW = cs (ω) dω.

2

a wave crest (in the form of a pure cosine; see ’2 1

middle curve in Figure 11.11) will, on average,

be replaced by a pure sine after a characteristic Thus, negative frequencies represent westward

time ζ (upper curve in Figure 11.11). When there travelling waves, and positive frequencies east-

are eastward propagating waves, a large positive ward travelling waves.22

cosine coef¬cient at a given time t (the middle

22 Note that this convention depends upon the de¬nitions of i)

curve, Figure 11.11, which is represented by

the sign of the quadrature spectrum, ii) the variance of eastward

ct = 1 and st = 0) will tend to be followed and westward travelling waves, iii) R, and iv) the frequency“

by a large positive sine coef¬cient ζ time units wavenumber spectrum. They are to some extent arbitrary.

latter (upper curve, represented by ct+ζ = 0 Negative frequencies are associated with westward propagation

and st+ζ = 1), and will tend to have been for the particular the eastwardused here. wave variance appears

de¬nitions An advantage of this

convention is that travelling

preceded by a large negative sine coef¬cient ζ on the right hand side of diagrams, and westward travelling

time units earlier (lower curve, represented by wave variance on the left (cf. Figure 11.12).

11.5: Frequency“Wavenumber Analysis 245

11.5.5 Examples. We return to the three

bivariate AR(1) examples that were discussed

Westward Eastward

in [11.4.7,8]. The general representation used here Travelling Travelling

Waves Waves

is

1 0

C C Zct

√

=A +σ .

St S Zst

0 b

t’1

The ¬rst example has a simple AR(1) cosine

process and a white noise sine process. Thus

± 0

A= .

0 0

We calculated the spectra needed to determine the

frequency“wavenumber spectrum in [11.4.7]. The Figure 11.12: Frequency“wavenumber spectra

fw

of a bivariate process with a rotational

power spectrum of C is a red spectrum (11.78), and cs

that of S is white (11.79). The quadrature spectrum parameter matrix. The same examples are shown

as in Figure 11.10, namely u = 0.95 and r =

is zero so the frequency“wavenumber spectrum,

0.99/0.50, and u = ’0.4 and r = 0.8/0.7. It is

σ2 1

fw

+ b , assumed that v is negative. The ˜theoretical peak

cs (ω) =

2 1+± 2 ’ 2± cos(2πω)

frequencies™ · = cos’1 (u) = 0.315 and 0.05

(11.92) are marked by thin vertical lines. Variance at

positive frequencies is interpreted as coming

is symmetric. Equal variance is attributed to

from eastward travelling waves, and variance

eastward and westward travelling features.

at negative frequencies from westward travelling

The second example has two unrelated red noise

waves. The right axis measures the spiky spectrum,

processes with the same parameter ± and forcing

whereas the left axis is valid for the other three

noise of the same variance. That is

spectra.

±0

A=

0±

Depending upon the sign of v, this process

and b = 1. The spectra needed to determine the exhibits a smooth transition either from a cosine

frequency“wavenumber spectrum were derived pattern to a sine pattern (eastward travelling

in [11.4.7]. Since the quadrature spectrum is zero, waves), or from a cosine pattern to an inverse

the frequency“wavenumber spectrum sine pattern (westward travelling waves). The

transition tends to occur in „ = 8π cos’1 (u) time

1

1

fw

cs (ω) = ( cc (ω) + ss (ω)) = cc (ω) steps. Sequence (11.83) shows that the eastward

2

motion (i.e., to the right) occurs when v is

is again symmetric. A preferred direction wave positive. Conversely, negative v is associated with

propagation is not indicated. sequence (11.84) and westward motion.

The third example, with a rotational parameter Figure 11.12 shows frequency“wavenumber

matrix spectra for examples with the combinations of

periods, represented by u, and damping rates r

u ’v

A=r , considered previously. This time we assume v is

v u

negative, so we can expect westward propagation.

is more interesting. In this case the frequency“ Indeed, we see that, except for the strongly damped

process (r = 0.5), most variance is ascribed to

wavenumber spectrum

westward travelling waves. The ˜theoretical™ peak

fw

cs (ω) = cc(ω) ’ cs (ω) frequencies · = 2π cos’1 (u) are associated with

1

σ2 maxima in the spectra.

= 2 1 + r2 (11.93)

D

’ 2r (u cos(2π ω) ’ v sin(2π ω)) 11.5.6 Example: A Two-sided Spectrum

of Travelling Wave Variance. Fraedrich and

is not symmetric.23 B¨ ttger [124] studied ¬ve years of daily Northern

o

23 See [11.4.8] for the power and quadrature spectra. D 2 is Hemisphere winter 500 hPa geostrophic merid-

ional wind data that was derived from daily

given by (11.82).

11: Parameters of Univariate and Bivariate Time Series

246

between ˜propagating™ and ˜standing™ wave vari-

ance by arguing that a standing wave is the sum

of two waves of equal variance that propagate in

opposite directions. Motivated by this reasoning,

Pratt partitions the total variance σT into two

2

symmetric spectra: one describing the distribution

of standing variance with time scale, the other

describing propagating variance. Since the spectra

are symmetric, they are de¬ned so that the total

variance is ascribed to positive frequencies.

The standing wave variance spectrum is de¬ned

as

fw fw

cs (ω) = 2 min cs (ω), cs (’ω) .

st

Figure 11.13: Two-sided spectra of travelling (11.94)

wave variance of daily meridional geostrophic

Note that cs is symmetric in ω.

st

wind during winter at 50—¦ N. The vertical

The propagating wave variance spectrum is

axis represents the zonal wavenumbers k =

de¬ned as

0, 1, . . . , 10. The time scale is given on the bottom

pr o p p