geopotential heights at various tropospheric levels represented in a semi-spectral form as

and a conventional linear stability analysis of

(θ, φ, z, t) = 1 (φ, z, t) cos(kθ)

the quasi-geostrophic vorticity equation were

+ 2 (φ, z, t) sin(kθ) (15.19)

compared. Both analyses are expected to detect

signals that propagate more or less zonally on where θ is longitude, φ is latitude, z represents

15.2: Examples 341

height, and k is the zonal wavenumber. Note that

(15.19) can be re-expressed as

(θ, φ, z, t) = Re 1 (φ, z, t) + i 2 (φ, z, t)

’ikθ

—e

= Re A(φ, z, t)e’i(kθ ’ (φ,z,t))

(15.20)

where A(φ, z, t) and (φ, z, t) are the amplitude

and phase of zonal wavenumber k. This represen-

tation will be used in the diagrams.

A separate POP analysis was performed for each

wavenumber k on the random vector composed

of trigonometric coef¬cients 1 and 2 of

geopotential height at all latitudes and heights. The 1%

system matrix A of (15.1) was estimated from 5%

10 %

winter (DJF) observations for 1984/85 through

1986/87.

The data were band-pass ¬ltered to remove

variability on time scales shorter than 3 days and

longer than 25 days. Also, the dimensionality of

the problem was reduced by using a truncated EOF

expansion. The ¬rst 18 EOFs, which represent

Figure 15.5: Baroclinic waves: Cross-spectral

more than 95% of the total variance for each

analysis of the POP coef¬cient time series shown

wavenumber, were retained.

in Figure 15.4. The vertical dashed line marks the

Here we discuss only the POP obtained for

POP period T . The horizontal dashed lines in the

Northern Hemisphere wavenumber 8. The POP

coherence plot (bottom) depict critical values for

represents 54% of the wavenumber 8 variance, has

tests of no coherence null hypothesis at the 10%,

a period of 4.0 days, and an e-folding time of time

5%, and 1% signi¬cance levels. From Schnur et

8.1 days. Note that the decay time is sensitive to

al. [341].

the type of time-¬lter.

Since the state vector Xt consists of the sine and

cosine coef¬cients of zonal wavenumber 8, both The maximum variance is found in the three-

the real and the imaginary part of the complex to ¬ve-day time scale, the phase difference is

POP, p = p r + i p i , must also be interpreted uniformly 90—¦ , as it should be, and the coherence

as vectors of sine and cosine coef¬cients. These, is high in the neighbourhood of the POP period of

in turn, can be represented as amplitude patterns four days.

Ar and Ai composed of amplitudes Ar (φ, z) and The system matrix A in (15.1) can also be

Ai (φ, z), respectively, and corresponding phase obtained from theoretical considerations. Schnur

patterns r and i . These patterns are shown in et al. [341] did this by using a standard

Figure 15.3 as height-latitudinal distributions. The perturbation analysis to linearize the quasi-

amplitude ¬elds Ar and Ai are almost identical, geostrophic vorticity equation on a sphere around

and the phase distribution r is shifted 90—¦ the observed zonally averaged mean winter state.

eastward relative to i at those latitudes where The linearized system was then discretized. The

the amplitudes are large. We therefore conclude resulting system equation for the streamfunction

that the diagnosed POP describes an eastward was expressed in the form of (15.1) by using

travelling pattern. representation (15.19) for the streamfunction for

The estimated coef¬cient time series z tr and z ti each wavenumber k and forming the (unknown)

vary coherently, with z tr lagging z ti by one or state vector X from 1 and 2 as above.

two days (Figure 15.4). This visual interpretation

The resulting system matrix A has complex

is substantiated by the cross-spectral analysis7

eigenvectors q = q r + i q i . The complex

of the two coef¬cient time series (Figure 15.5).

eigenvalue that is connected with the pattern q can

be written as » = ξ e’i· , where T = 2π/· is the

7 Spectral and cross-spectral estimation techniques are

period of a cyclical sequence like (15.8) involving

described in Sections 12.3 and 12.5.

15: POP Analysis

342

the real and imaginary parts of q , and where

Phase

the value of ξ determines whether the system hPa

ampli¬es or damps these oscillations. Thus, as

with POP analysis, the normal modes represent

propagating waves. The phase direction depends

on the eigenvalue.

However, there are also important differences

between the POP and perturbation analysis

techniques. We mentioned that POP analysis of

stationary data yields eigenvalues |»| < 1.

POP analysis based on the estimated matrix A North Latitude

preferentially ˜sees™ oscillations in their mature

state (i.e., when noise is comparatively small

Amplitude

and when there is damping by nonlinear and hPa

other processes). In contrast, the system matrix A

obtained from perturbation analysis describes the

early evolution of small deviations from a speci¬ed

basic state. This system will amplify many of

these initial perturbations, and these are in fact

the solutions that are of interest. Thus it is the

modes with eigenvalues |»| ≥ 1 that describe

the growing oscillations that the POP analysis

eventually detects.

North Latitude

Just as with POPs, both q r and q i can be

represented by amplitude and phase patterns.

Figure 15.6: Baroclinic waves: The amplitude and

However, since the system matrix depends only on

phase of the fastest growing Northern Hemisphere

a zonally averaged basic state, the solutions must

zonal wavenumber 8 normal mode. The mode

be invariant with respect to zonal rotation (unlike

was obtained from a perturbation analysis of the

the POPs). It can therefore be shown that q r and

discretized quasi-geostrophic vorticity equation

q i have equal amplitude and that the phase of q i

is just that of q r shifted by ’90—¦ . That is, q i is linearized about the observed zonal mean state in

Northern winter. The amplitude grows e-fold in 2.2

redundant.

days, and the period is 3.9 days. From Schnur et al.

The most unstable normal mode (i.e., with the

[341].

greatest eigenvalue |»| ≥ 1) obtained for Northern

Hemisphere wavenumber 8 has a period of 3.9

days. This is an eastward propagating growing

atmosphere with similar oscillation period: the

mode that increases amplitude e-fold in 2.2 days.

stratospheric Quasi-Biennial Oscillation (QBO)

The amplitude pattern Ar (Figure 15.6) of this

and the tropospheric Southern Oscillation (SO).

normal mode is almost identical to the amplitude

The QBO can be observed in the stratospheric

patterns of the POP shown in Figure 15.3. The

equatorial zonal wind with time series available

normal mode has a large maximum near the

surface at 40—¦ N because the perturbation analysis

at six stratospheric levels. POP analysis was

performed on deviations from the long-term mean.

did not account for friction. The phase pattern r

differs from the POP phases i = r ’ π/2 by No time-¬ltering was done for this data set.

only a constant angle. Monthly mean anomalies of the 10 m zonal

wind along the equator between 50—¦ E and 80—¦ W

In summary, POP analysis, which estimates

and of the equatorial sea-surface temperature

the system matrix from observations, ¬nds