cc (ω) ’ ss (ω)) + 4( cs (ω)

2

C 2 (ω) = .

2

cc (ω) + ss (ω)) ’ 4( (ω)

2

cs

(11.98)

The propagating variance is then de¬ned as the

remainder

1

fw

pr o

cs (ω) = cs (ω) ’ cs (|ω|).

st

(11.99)

2

The total standing wave variance is de¬ned as the

integral of the standing wave spectrum over the

positive frequencies:

1

2

σst = cs (ω) dω

2 st

(11.100)

0

In contrast, the total propagating wave variance

is de¬ned as the integral of the propagating wave

spectrum over all frequencies:

1

2 pr o

σ pr o = cs (ω) dω.

2

(11.101)

’1

2

Hayashi also devised a method for ascribing a

(spatial) phase to the standing wave variance at Figure 11.15: One-sided spectra of standing and

frequency ω. For wavenumber k, the position of propagating wave variance (Pratt™s de¬nition) of

500 hPa geopotential height during winter at 50—¦ N

the train of crests and troughs of this standing

plotted as a function of zonal wavenumbers k =

wavenumber relative to the origin is given by

1, . . . , 10 in the vertical and log ˜periods™ in the

2 cs (ω)

1 ’1

φcs (ω) =

k horizontal.

tan

cc (ω) ’ ss (ω)

2k Top: Propagating wave variance. Shading indi-

when both the numerator and the denomina- cates westward propagation.

tor are nonzero. When both are zero, the ˜co- Bottom: Standing wave variance.

herency™ C(ω) (11.98) and the standing wave From Fraedrich and B¨ ttger [124]. o

variance are also zero, so the phase is meaningless.

When the denominator is zero (i.e., cc (ω) =

ss (ω)) the phase is given by

We will see in the example below that Hayashi™s

method generally gives useful results. However,

π

4k if x y (ω) > 0

from a strictly mathematical point of view, the

φcs (ω) =

k

formalism is a not entirely satisfactory because it is

π

’ 4k if x y (ω) < 0

sometimes possible to obtain negative propagating

and when the numerator is zero (i.e., when spectral densities (11.99).

x y (ω) = 0), it is given by

cc (ω) > ss (ω)

0 if 11.5.11 Luksch™s Examples Revisited. To

φcs (ω) =

k

π

cc (ω) < ss (ω).

illustrate Hayashi™s formalism we return once

if

2k

more to the three examples described in [11.4.7,8],

See [170] for details. [11.5.5], and [11.5.8].

11.5: Frequency“Wavenumber Analysis 249

spectrum is an even function of frequency. Thus, in

In the ¬rst example, the cosine coef¬cient

keeping with the nature of the parameter matrix A,

evolves as an AR(1) process with parameter ± and

forcing with variance σ 2 , and the sine coef¬cient variance has no preferred direction of propagation.

evolves as white noise with variance bσ 2 . Then The second example consisted of cosine and

(cf. (11.92)) sine coef¬cient processes generated by two in-

dependent AR(1) processes with identical AR

σ2 1

fw

cs (ω) = +b parameter and variance. Thus, the two spectra cc

2 D2 and ss are equal and the ˜coherency™ C(ω)

|1 ’ bD 2 | vanishes so that the total variance is distributed

C(ω) = ,

1 + bD 2 equally among the westward and eastward prop-

with D 2 = 1+± 2 ’2± cos(2π ω). D 2 is symmetric agating waves at all frequencies, as

in ω so that fw

pr o

cs (ω) = cs (ω). (11.102)

σ 2

fw

cs (ω) = cs (ω)C(ω) = |1 ’ bD |.

st 2

2 D2 The third example used a bivariate AR(1) pro-

The distribution of the total variance to cess with a rotation matrix as its parameter matrix

the standing wave, eastward, and westward A. In the setup considered in [11.4.7], [11.5.5],

propagating components depends on the ra- and [11.5.8], the white noise forcing parameter b

tio cc (ω)/ ss (ω) = bD 2 . was set to 1, resulting in cosine and sine coef¬cient

When cc (ω) = σ 2 /D 2 is greater than processes of the same variance. In this case all

ss (ω) = σ b, that is, if bD is less than 1, then

2 2 of the variance is again attributed to propagating

waves (11.102).

σ2

cs (ω) = (1 ’ bD 2 )

st When b is not 1, the variances of the cosine and

2

2D sine coef¬cient processes are not equal and part

pr o

cs (ω) = bσ

2

of the joint variance is attributed to standing wave

φcs (ω) = 0.

k variance. In this case the frequency“wavenumber

spectrum (11.93) is given by (see Luksch et

The cosine series dominates the sine series in this al. [262] for details)

case, so setting the phase of the standing waves

to zero is reasonable. In the limit, when b = (1 + b)σ 2

fw

cs (ω) =

0, all the variance is attributed to standing wave 2D 2

variance. Also, note that when the standing and

— 1 + r 2 ’ 2r (u cos(2π ω) ’ v sin(2π ω)) ,

propagating spectra are integrated, as in equa-

tions (11.100, 11.101), the total propagating and

standing variance sums to σT . In the opposite case, and the squared ˜coherency™ is

2

with the cosine coef¬cient spectrum smaller than 1’b 2

C(ω)2 =

2 is greater

the sine coef¬cient spectrum, that is, bD