axis. The top axis gives the associated phase

velocity (see text). where

From Fraedrich and B¨ ttger [124].

o 1 st

fw

p

cs (ω) = cs (ω) ’ (ω).

2 cs

geopotential analyses. They ¬rst calculated the This spectrum is also symmetric.

We can express these spectra in terms of

cosine and sine coef¬cients of the meridional wind

—¦ N for spatial wavenumbers k, k = 0, . . . , 10. the power and quadrature spectra of the cosine

at 50

Two-sided spectra of the travelling wave vari- and sine coef¬cient processes by substituting

ance (11.91) (Figure 11.13) were estimated from de¬nition (11.91) of the frequency“wavenumber

spectrum into (11.94) and (11.95). We ¬nd that the

the cosine and sine coef¬cient time series.24

A few words are required about the presentation standing and propagating variance spectra can be

in Figure 11.13. First, continuous contours are expressed as

cs (ω) = cc (ω) + ss (ω) ’ 2| cs (ω)| (11.96)

st

used for clarity even though the frequency“

wavenumber spectrum is discrete in k. Also, for

and

each wavenumber k, period 1/ω can be interpreted

pr o

cs (ω) = 2| cs (ω)| (11.97)

as a phase velocity that expresses the rate at which

the wave crest moves. The lines of constant phase respectively.

velocities are indicated by tick marks on the upper Pratt uses the frequency“wavenumber spectrum

axis. to label the propagating variance as eastward or

Most of the variance is attributed to zonal westward. If the frequency“wavenumber spectrum

wavenumbers k = 2, . . . , 7. The variance at assigns more variance to eastward than westward

large scales (wavenumbers k = 2, 3) is divided travelling waves at a given frequency, the

equally between eastward and westward travelling propagating variance at that frequency is labelled

waves. Almost all smaller scale variability that is ˜eastward,™ and vice versa. Equivalently, if the

characteristic for baroclinic dynamics (k ≥ 5) is quadrature spectrum at a given frequency is

attributed to eastward travelling waves. A variance positive, then the propagating variance in the

maximum occurs along a line that corresponds neighbourhood of that frequency is identi¬ed as

well with the theoretical dispersion line for Rossby eastward.

waves at 50—¦ N in a zonal mean ¬‚ow of about The total standing wave variance and the

15 m/s [56]. total propagating wave variance are the integrals

over all positive frequencies of the standing and

11.5.7 Pratt™s De¬nition of Standing Wave Var- propagating variance spectra:

iance Spectra. Pratt [319] tries to discriminate 1

2

σst = cs (ω) dω

2 st

24These spectral estimates are subject to uncertainty from a 0

number of sources, whose effects we ignore for the moment. 1

2

Spectral estimation is discussed in some detail in Sections 12.3 pr o

σ pr o = cs (ω) dω.

2

and 12.5.

0

11.5: Frequency“Wavenumber Analysis 247

By substituting equations (11.96) and (11.97) into

these expressions, we see that the sum of the u = 0.95

standing and propagating wave variance is the total

variance (11.88)

σT = σst + σ pr o .

2 2 2

While Pratt™s partitioning of variability is

intuitively pleasing, we should remember that it is

based on heuristic arguments. Therefore, as with

other aspects of the language used in frequency“

wavenumber analysis, the terms ˜standing and

propagating wave variance™ are an ambiguous

description of equations (11.94) and (11.95). u = -0.4

Literal interpretation of these quantities as the

standing and propagating variance spectra can be

misleading.

11.5.8 Examples of Pratt™s Decomposition.

Recall again the three bivariate AR(1) exam-

ples developed by Ute Luksch (see [11.4.7,8]

and [11.5.5]). Since the quadrature spectrum is

zero in the ¬rst two examples, Pratt™s formalism

attributes all variance to standing waves. This

Figure 11.14: Standing (dashed) and propagat-

clearly makes sense in the ¬rst example because

ing (solid) variance spectra for a bivariate pro-

the cosine coef¬cient varies dynamically and the

cess with rotational parameter matrix (cf. Fig-

sine coef¬cient is white noise. Interpretation is a

ure 11.12). The ˜theoretical™ peak frequency · =

little more dif¬cult in the second example where

’1

2π cos (u) is represented by a vertical line.

1

cosine and sine coef¬cients are independent, iden-

Top: u = 0.95; r = 0.99 (left hand axis, the

tically structured AR(1) processes.

propagating wave spectrum is scaled by a factor

The propagating and standing wave variance

of 0.01) and r = 0.5 (right hand axis).

spectra for Luksch™s third example, in which

Bottom: u = ’0.4; r = 0.8, and r = 0.7. The

the bivariate AR(1) processes have rotational

standing wave spectra can not be distinguished in

parameter matrices, are shown in Figure 11.14.

this diagram.

In all cases, most of the variance is attributed to

the propagating variance and only a small portion

is designated as standing variance. Except for

the u = 0.95, r = 0.5 process, the peaks in was estimated to occur on time scales longer

than 10 days with maximum variance for small

the propagating spectra correspond well with the

wavenumbers (Figure 11.15, top). The bulk of the

theoretical rotation rate 2π/· (· is indicated by

variance due to propagating waves, on the other

the vertical line in the diagrams). As with the

hand, was attributed to time scales of less than

coherency spectrum (see Figure 11.10), the peak in

the propagating spectrum when u = 0.95, r = 0.5 10 days and baroclinic spatial scales. Almost all

occurs at a frequency greater than ·. variance was attributed to eastward propagating

Note that negative values of v were used variance (Figure 11.15, bottom). Fraedrich and

to compute Figure 11.14. Negative v results B¨ ttger were able to relate dynamically the three

o

in negative quadrature spectra (for positive ωs) spectral maxima in Figure 11.13 to standing waves

or eastward propagating waves.

so that all propagating variance is attributed to

westward travelling waves.

11.5.10 Hayashi™s De¬nition of Standing Wave

11.5.9 Example [11.5.6] Revisited. Fraedrich

Variance. An alternative to Pratt™s approach was

and B¨ ttger [124] applied Pratt™s formalism to

o

offered by Hayashi [170], who de¬nes a non-

time series of daily analysed 500 hPa geopotential

height during winter along the 50—¦ N latitude symmetric spectrum for propagating variance, and

circle. Most of the standing wave variance a symmetric spectrum for standing variance.

11: Parameters of Univariate and Bivariate Time Series

248

In this formalism, the standing wave spectrum is

de¬ned to be

fw fw

cs (ω) = C(ω) cs (ω) cs (’ω)

st

with ˜coherency™