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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 89, NO. D5, PAGES 7202-7214, AUGUST 20, 1984

Baroclinic Generation of Planetary Transient and Stationary Waves From Forced Stationary Waves

J. DAVID NEELIN

Geophysical Fluid Dynamics Laboratory, Princeton University

CHARLES A. LIN

Department of Physics, University of Toronto

The linear instability of forced stationary waves in a barodinic zonal flow is examined with a two-level quasi-geostrophic beta plane model. Realistic zonal topography and diabatic forcings produce a steady state solution of planetary-scale stationary waves. Baroclinic instability of the forced waves gives rise to transient perturbation modes of planetary zonal scale and a preferred meridional scale of about twice the radius of deformation. One of the dominant planetary modes is stationary and resembles the observed stationary wave pattern. In the steady state model this form can be obtained only for certain values of the meridional scale of the forcings. It is suggested that this stationary mode could contribute to the time average stationary wave distribution. Applications to blocking are also considered. Interference of stationary and quasi-stationary modes can apparently produce regional as well as global blocking patterns.

1. INTRODUCTION

Planetary scale waves comprise a major portion of both stationary and transient eddy statistics in the northern hemisphere ['dulian et al., 1970; Kao and Sa•?endorf, 1970; Willson, 1975; Blackmort, 1976]. The generation of transient wave energy at these scales is not well explained by the zonally symmetric baroclinic instability problem that has provided a successful model for the development of synoptic scale disturbances since the work of Charney [1947] and Eady [1949]. For stationary waves, on the other hand, the importance of planetary scales has long been understood in terms of the zonally asymmetric, stationary forcing of the flow. Both topography and the diabatic heating associated with land-ocean contrast are dominated by wave numbers 1, 2, and 3 [Derome and Wiin-Nielsen, 1971]. A simple approach to modeling the stationary waves is to view the time mean circulation as approximately the steady state response of the flow to these forcings. Beginning with Smagorinsky [1953], various investigators have shown that these stationary equilibrium models can reproduce some features of the observed stationary waves. However, the distinction between stationary and transient waves that is implicit in this approach may not be entirely appropriate. It is possible that the forcings maintaining the stationary waves also contribute to the transient planetary wave variance. Conversely, transient motions can potentially contribute to the time average stationary wave pattern.

In order to investigate the relationship of time-dependent motions to the zonally asymmetric forcings and stationary waves, we consider small perturbations to a forced wave stationary equilibrium. To obtain the basic state, we follow a model developed by Derome and Wiin-Nielsen [1971] that includes both topographic and diabatic forcings estimated from observation. Linear instability of the forced waves gener-

Copyright 1984 by the American Geophysical Union.

Paper number 4D0467. 0148-0227/84/004D-0467505.00

ates growing perturbation modes, which we examine as an indication of the time-dependent evolution of the flow near the steady state solution. Both traveling and stationary planetary wave modes arise, growing baroclinically from the forced wave available potential energy. Our results apply to three major questions.

First, we suggest that planetary, zonally asymmetric forcings can indeed provide for transient planetary waves. Lin [1980a, b] has previously hypothesized this by analogy with baroclinic instability of a free Rossby wave, and Sasamori and Youngblut [1981] have done the same on the basis of instability of a simple, prescribed stationary wave to perturbations of highly restricted form.

Second, we find a growing stationary mode which suggests that the time-dependent motions may affect the form of the time average stationary wave pattern. A distribution of shorter meridional scale will tend to grow from a forced solution of long meridional scale.

The third possible application is to blocking. In this area our work complements that of Frederiksen [1982]. He exam- ines the linear instability of an observed winter stationary wave pattern and zonal flow in a two-level model with spherical geometry. For low static stability he finds cyclone-scale monopole structures to be the fastest growing modes, whereas at greater stability, slowly moving dipole modes of longer zonal scale dominate. Dipole refers to the north-south structure of the mode, indicating that high-low vortex pairs are found. Frederiksen qualitatively identifies these dipole modes with blocking configurations and invokes the modon theory of blocking [McWilliams, 1980, and references therein] to hypothesize that the phase speed of the modes would be reduced as they grew to finite amplitude. In our study, explicit inclusion of the stationary forcings allows us to incorporate the direct interaction of the growing perturbation with topography. This produces planetary-scale modes with significant growth rate and tends to reduce the phase speed of long and planetary modes, rendering certain of them quasi-stationary or stationary. We tentatively associated these modes with

7202

NEELIN AND LIN' BAROCLINIC GENERATION OF PLANETARY WAVES 7203

components of observed blocking patterns and show that they can potentially account for both regional and double blocking. We also examine the meridional-scale selection process, which produces the favored dipole modes.

2. THE MODEL

2a. Two-Level Quasi-Geostrophic Model

The model used is the well-known two-level quasi- geostrophic model on a mid-latitude beta plane [Phillips, 1954]. We use the notation listed in the section at the end of the paper.

The two-level model system is that found in Haltiner and Williams [1980] but with diabatic heating included in the thermodynamic equation and with nonzero vertical velocity at the lower boundary'

• V2½ -- --J(½, V2½ q-f)- J(q•r, V2½T) q- 2'-•p (DB (1) 3t

3• (V2 - •2)•r = -J(•' (V2 - •2)•r)- J(•r, V2• + f) fo

The diabatic heating field Q is specified from observations. We obtain the vertical velocity forced by topography at the lower boundary by

h Ap w• = -p•. V 2H = - • J(• - •r, h) (2)

The wind at 750 mbar is used in (2) for energetic consistency. The beta-plane assumption and rigid lid boundary con-

dition can be expected to compromise the results to some extent, es•cially at the planetary scale [Kirkwood and Derome, 1977; Lindzen et al., 1968]. These are frequently adopted for the dramatic simplification they bring to the problem. Moreover, this formulation allows us to make use of an existing steady state model of the forced stationary waves' that of Derome and Wiin-Nielsen [1971]. Although we do not expect our results with this model to •ve a quantitatively accurate picture, we do expect them to reflect qualitatively certain dynamic processes that are important in the maintenance of both stationary and transient planetary waves.

2b. Steady State Model

We are concerned with perturbations to a stationary equilibrium solution to system (1). We choose as our basic state a modified version of the model by Derome and Wiin-Nielsen [1971, hereafter referred to as DW]. This model includes both diabatic and topographic forcings that are assumed to have the form

M

[h, Q] = • [h•, Q•] exp {imkx} cos (•y/Ar) (3) m= --M

-}AY•y•}AY

Since in the DW model the motion is restricted to this latitude

band, the half wavelength A Y is a suitable measure of meridional scale. In our instability analysis we continue to use the perturbation half wavelength A Y' as a convenient measure, even though there are no lateral restrictions on the motion.

In order to obtain exact rather than linearized solutions for

our basic state we assume that the north-south extent of the

forcing is much larger than the region of dynamic interest and thus make the approximation A Y• oo. There then exist steady state solutions to system (1) of the form

M

[W, WT] = -[u, ur]y + • [am, am T] exp {imkx} (4) m----- --tiff'

where the stationary wave components {am, am r} are given by 2M separated algebraic systems as described in DW.

2c. Time-Dependent Perturbation Equations

We write the stream function as the sum of the steady state solution plus a transient perturbation [½', ½r'], and we lin- earize about the stationary equilibrium to obtain

V2½'= -J(W, V2½ ') - J(½', V2W +f)- J(Wr, V2½r ')

- v'-%) - « foJ(½' - (5) (v: = -J(,I,, (v'- J(½', (v: -

- J(Wr, ) - J(½r', V2W +f) + «foJ ½'- ½r, (5')

We then express the perturbation as a Fourier expansion with general spatial dependence:

[½' ½r'] = YnI)] exp {i(nkx + lv- at)} dl

(6)

Note that while the beta plane is zonally periodic we do not impose any walls in the meridional direction, requiring only that the solutions be bounded as y--, + oo. We note that the perturbation is expanded in harmonics of the fundamental zonal wave number k rather than in multiples of a basic state wave number, as has been the case in some previous studies [e.g., Sasamori and Younoblut, 1981].

The problem is separable in the meridional wave number I. We thus obtain the following nondimensionalized eigenvalue problem, which can be solved independently at each value of l'

(%- it)X.+ d..+

f,,+ (% - it)Y,+

M

m= -M

M

E (tnmSn+m q- VnmL+m)= 0

(7)

for --N _< n _< N, 0 _< I < oo, and where

C n -'- n[1 - U- l(n2K2 + L2) - 1]

d n = nU/Ur

f.= -nU/UT[1 -(n2K 2 + L2)]/[1 + (n2K 2 + L2)]

e.= n[(1 - U- •(1 + n2K 2 + L2) - 1]

rn m= --imL[(l+ 2nmK2\

Sn m --' -- imL 1 + n2 • T •L2 A_m T

7204 NEELIN AND LIN.' BAROCLINIC GENERATION OF PLANETARY WAVES

t. ==-imL 1-iT•/•._7_L •' A-•r (8)

- 'I+n:K:+L a 2•o'

v• •= -imL 1 + 1 +n•• L a' A_•

+ 'I+n:K:+L a 2• o' The following notation has been used for the nondimensionalized variables'

=

EK, L] = Ek, 1•/•

Ro = •U/fo

EA•, A•r• = Ea•, a•r•/u

Solving system (7) over all positive values of 1 provides complete generality, since the transformation 1• - 1, n • - n leaves the eigensolutions unchanged.

When the stationary wave amplitudes and topography are zero, the interaction terms r• •, s• •, t• •, v• • in system (7) disap- pear, and the problem separates in the zonal Fourier series as well. This leaves 2N + 1 two-component eigensystems corresponding to the classical problem of baroclinic instability of a zonally symmetric baroclinic flow. For this problem there is a critical value of the zonal thermal wind Ur•nt = •/•:, below which no instability occurs at any wavelength. In our model we use a radius of deformation •- • = •.1• x 106m, and thus this critical shear value is Ur•nt = 4.3 m/s. We use the terms

1607

o-

-•4o.- • / -•2o• •//

] , I • I , I , I , I , I , I , I ,

0

-240

,

-120

0 ø 40 80 120 160 160 120 80 40

EAST LONGITUDE WEST

Fig. 1. Stationary wave geopotential height at three levels. Best fitting results from Derome and Wiin-Nielsen [1971] steady state model (broken curve) and observed pattern for the relevant winter period (solid curve).

24o / 25cb •'.•-,-..• I .. , 160[-- •\ / " •-.-/ .,.,Ix. x i \ --'x -/ /"' ø'"

-8o i / '- -',,,: / -160 -

- ".X, -:320 - , I I I , I I I , I , I , I , I ,

160 - 50cb

= LU -80- \ N \ /

-240 - , I , I , I , I , I , I , I , I ,

12Or- _ _ 75cb /'•.

/ -,oF'-.-.'" -120

, I , I , I .-•d' O* 40 80 120 160 160 120 80 40 0

EAST LONGITUDE WEST

Fig. 2. DW model statioriary wave height at three levels for A¾ = 23 ø (solid curves), 30 ø (dashed), arid 50 ø (dot-dashed) oœ latitude (figure [rom Dcrom½ and Wii.-Ni½ls½. [i9?l]).

"classical" or "conventional" to refer to the baroclinic insta-

bility process found in this problem and its vertically continuous analog.

2el. Basic State

Using the basic state model described in section 2b, we prescribe diabatic heating and topography fields that are realistic in the zonal direction. We take the values used by DW. The only free parameters besides the radius of deformation are the basic state mean and thermal zonal wind, since k, fo, and fi are determined by our choice Cpo = 45 ø. In the main case presented in this paper we set ur = ttTerit, the critical shear wind, and u = 3.0 x ttTeri t = 13 m/s. Both of these were subsequently varied within climatological bounds to test the sensitivity of the results. Our choice of the thermal wind for this main case

is a very reasonable one, since it has been pointed out by Stone [1978] that the observed average tropospheric zonal wind gradients coincide remarkably close to critical shear in mid and high latitudes.

We now present some of the results of the DW version of the steady state mode and compare our basic state to these. In Figure 1 the observed meridionally averaged stationary wave heights are shown along with the best fitting model results as presented by DW. This fit is obtained by setting the north- south extent of the mountain and heating forcing to A Y = 30 ø of latitude. However, the response is quite sensitive to the choice of this parameter. Figure 2 gives an indication of this sensitivity, showing the different patterns obtained for A Y = 23 ø, 30 ø, and 50 ø of latitude. The susceptibility of the response to variation of the forcing meridional wave number provides additional motivation for examining the stability of the long wavelength limit.

In Figure 3 we see the basic state used in most of the rest of this study. Because we have simply set A Y = oo, the model response is quite dissimilar to the best fitting case and is out of phase with the observed pattern. It resembles the A Y = 50 ø of

NEELIN AND LIN' BAROCLINIC GENERATION OF PLANETARY WAVES 7205

2,;0[ .... ' ............. -• I\

•6o1• / • J

-80F \ x / • \ / -I -16o • -240 - •.? - -:520

160

80 o

-80 -I• -240

40

-40

-I 20

- 250mb - I I I I I I I I I I I I i I i I

'=•-'-, - \ i I \•.•/! - - \•/I -

, I , I , I I I i I i I i I • 1

?50rob -

, •., •, • ,•, • , • , •, 0 ø 40 80 I 0 I 0 160 120 80 40 0 ø EAST WEST

LONGITUDE

Fig. 3. Stationary wave height at three levels. Basic state used in principal case of perturbation analysis (solid curve) and observed pattern (dashed curve).

latitude case if one allows for the fact that DW included forc-

ings at zonal wave number 4 and higher. We note that there are small differences in the parameters

used in our basic state model as compared to the DW model but that these only have minor effects. The DW model incor- porates damping parameterized as a lower-boundary vertical velocity proportional to vorticity. For long meridional wave- lengths this parameterization does not have an appreciable effect, and so we have simply dropped it for clarity. Because we do not attempt to include the effects of the relatively small forcing at zonal wave numbers of 4 and higher, we do not have to contend with the problem of resonance as DW did.

3. INSTABILITY ANALYSIS ß PRINCIPAL CASE

3a. Preferred Meridional Scale

In this section we examine the instability of the basic state described in section 2d and present the results of this principal case. In section 4 we then consider variations of the basic state

and find that the main case results are only slightly modified ß

•n most respects. The time-dependent perturbations we are considering are

initially permitted general spatial dependence as described by the Fourier expansion of (6). Solving the eigensystem (7), we find that the growth rate of the unstable normal modes varies markedly with their meridional wave number I. In Figure 4 we have plotted, as a function of l, the growth rate of the most unstable mode at each value. The sharp peak in growth rate around l = 0.5# indicates a preferred meridional scale for the perturbation.

The inset of Figure 4 focuses on this peak region, showing individual growth rate curves labeled A and B for the two dominant modes that share the peak. We can trace individual modes of the zonal eigensystern over a large range of I in this manner by their characteristic zonal Fourier spectrum and phase speed properties. Certain of the other unstable modes have sufficiently high growth rates that may be significant and will be described briefly. All of the significantly unstable modes in this principal case are planetary wave modes. Synop- tic scale modes are only marginally unstable as the basic state zonal wind is at critical shear and the stationary wave insta-

bility is inefficient at these scales. The reason for this and the mechanism of the meridional scale selection process will be examined in section 4b.

The peak growth rates of the "A" and "B" modes in Figure 4 correspond to e folding times of 9.2 and 8.7 days, respectively. This time scale is reasonable for the growth of planetary waves and is consistent with the time constant of dissi- pation of stationary waves by transient waves according to Sasarnori and Youngblut [1981]. The growth rate of these modes changes with variations of the basic state, although the essential structure and phase speed properties remain the same. We obtain e folding times down to 5 of 6 days for climatologically reasonable changes in the basic state.

The meridional scale that arises in this analysis corresponds to a perturbation half wavelength of A Y' = 29 ø of latitude. In spherical geometry this would be consistent with a meridional structure with (r/- v)= 4, where r/ and v are the total and zonal wave numbers of the associated Legendre function P,'(sin ½). It would also be consistent with (r/- v)= 5, but hemispheric geopotential height is commonly decomposed by using the even harmonics [Blackrnon, 1976]. We make this association between beta plane and spherical meridional structure by considering the local half wavelength at mid- latitudes of P,'(sin ½) [Mathematical Tables Project, 1945]. In this range of latitude we can expect flow distortion on the sphere relative to the beta plane to be small.

Our results suggest that components with this meridional wavelength should be important in low-frequency variance of planetary zonal scale. We can compare this with a spectral study of 10 years of northern hemisphere 500-mbar height data by Blackrnon [1976]. The power spectra for the winter season, which he has partitioned in intervals of 1/15 day -•, are strongly dominated by frequencies f= 0/15 day -• and f= 1/15 day -•. The spherical harmonic decomposition of power at f = 1/15 day-• has principal components (r/, v) = (5, 1), (6, 2), and (7, 3); in other words, zonal wave numbers 1, 2, and 3 with meridional structure given by (r/- v) = 4. Power at f= 0/15 day- • appears to be dominated by these same components. The scales arising from our instability analysis are thus consistent with observations.

.15

.125

.075

.05 -

.025 -

0.0 0.0

.40 .50 .60

.75 1.0 I. 25 ] , , [

.25 .50 1.50

MERIDIoNAL WAVENuMBER

Fig. 4. Growth rate of most unstable modes versus meridional wave number. Growth rate curves for the two dominant modes are

indicated in the inset. Meridional wave number in units of #- 1.94 x 10-am - • = (radius of deformation)- •.

7206 NEELIN AND LIN' BAROCLINIC GENERATION OF PLANETARY WAVES

3b. Dominant Unstable Modes

(1) Stationary planetary mode. Examining the two dominant unstable modes indicated in the inset of Figure 4, we find that the one labeled "A" has some very interesting properties. This planetary wave mode is essentially stationary and resembles the observed stationary wave pattern, as shown in Figure 5. This resemblance is remarkable in that the meridional scale

of the growing stationary wave is self-selected by the dynamics and is not an adjustable parameter as in the DW model. However, the scale of A Y'= 29 ø that arises is essentially the same meridional extent as DW found was necessary to fit their results to the observed pattern.

The mode is stationary in the zonal direction, as may be see from its Fourier spectrum, which is symmetric (Figure 6). In the meridional direction the pattern does, in fact, propagate, but at a negligible phase speed of 0.09 m/s. If the basic state forcing were simplified to just the preponderant wave number 2 elements, the phase speed would be exactly zero. The shape of the spectrum reflects the processes by which the mode is maintained, and we shall elaborate on these in section 4c. The basic state stationary waves and topography cause linking among the perturbation components. The dominant elements, wave numbers n = +__ 2 in the mean stream function and n = 0 in the thermal stream function, are linked by interactions with the wave number 2 components, which dominate the basic state.

Although we pointed out in Figure 5 the resemblance of this stationary mode to the observed stationary wave pattern, ob- viously we cannot emphasize this similarity without qualifi- cation. For one thing the amplitude of the wave is left arbitrary in linear theory, and so only the relative amplitude and phase of the pattern can be compared. Also, one must consider that the additional contributions of the basic state and the

time average of other growing modes may alter the picture. What is noteworthy is that we have reproduced a stationary, although growing, pattern which is extremely similar to that produced by the steady state model when AY is set to 30 ø of latitude. However, the meridional length scale A Y'= 29 ø is selected dynamically rather than specified exogenously. Hence not only will a forcing of meridional scale around 30 ø of latitude produce a stationary wave pattern similar to that ob-

L .......... ;, ...... _i I \

..... I \ • / • //

• 500rob 3

J I i I I I I I I I I I I I I I 0 ø 40 80 lEO I$O I• lEO 80 40 0 ø EAST WEST

LO•$1TUD•

Fig. 5. Stationary planetary wavemode ("A" mode) perturbation height (solid curve) scaled for comparison with observed wave (dashed).

-2 02 0

n n

(a) (b) Fig. 6. Perturbation zonal Fourier spectrum for the "A" mode, n is

zonal wave number.

served but forcing at a much longer scale is capable of producing a time mean field of exactly the same form. Consequently, one may speculate that were forcing simultaneously present at both scales the time average stationary wave field would con- tain components resulting from both direct steady state response and superposition of time-dependent eddies that re- peatedly grow with fixed zonal phase.

An ambiguity we cannot resolve in the context of our linear model is the selection of meridional phase. As it stands the stationary mode is unaffected by translation of the latitude 'origin, and therefore, in choosing between half-wavelength intervals, we have had a free choice of the sign of the mode. The phase is set by initial conditions and does not change significantly with time. In proposing that an ensemble of repetitions of stationary mode growth would contribute a statistical component to the stationary wave pattern we have made the implicit assumption that the phase of the mode is not initiated completely at random. Presumably, slow meridional variation of the forcings or finite amplitude interactions with the basic state would promote coherence in the meridional phase of the excited ensemble. This assumption is reasonable and is supported by results obtained by Yao [1980]. He employs a similar two-level model with mountain forcing at a single zonal wave number. The meridional half wavelength of the mountain is A Y = 60 ø of latitude, and he restricts the motion to this channel. Both the steady state solution and the long-term average of a time integration of the model are examined. In certain cases the two patterns are found to be highly dissimilar. The time average is dominated by components of the same zonal wave number as the forcing but with shorter meridional scale, which enables them to draw energy from the forced waves. These results are very similar to the picture we have suggested and imply that a nonzero ensemble average can exist regardless of the phase ambiguity of linear theory.

We consider further implications of this growing stationary mode in addressing applications to blocking in section 5.

(2) Slowly moving planetary modes. As may be seen from the spectrum in Figure 7 the other dominant mode (labeled "B" in Figure 4) is primarily composed of traveling plane wavesrathe asymmetric partmwhile the zonally stationary symmetric part is small. The spatial distribution of the mode is most readily understood when presented as a latitude- longitude plot, as in Figure 8, which shows contours of the 500-mbar height field. Interference of the plane wave components produces localized features. The 58 ø wide latitude band shown corresponds to one full cycle of the mode. To under- stand the time dependence of the distribution, it is sufficient to recognize that the pattern shown will be simply translated in latitude and scaled in amplitude by exp (a• t).

NEELIN AND LI•: BAROCLINIC GENERATION OF PLANETARY WAVES 7207

-$ o -I o

n n

(a) (b)

Fig. 7. As in Figure 6 but for "B" mode.

Although the components of the mode do travel, their phase speeds are fairly slow. In fact we can call this mode "quasi- stationary" in the following sense: when the ratio of growth rate to angular frequency, r = is unity, each wave component moves only 1/9 of a wavelength in the time it takes to double in amplitude. Thus it is appropriate to term growing waves as quasi-stationary when r •> 1, since they will grow to maturity before the phase changes sufficiently to produce can- cellation in the time average. For this case the e folding time of the mode is 8.7 days at peak growth rate. The resulting ratio, r-- 1.0, places it on the boundary we have drawn between transient and quasi-stationary waves. Thus at Peak growth rate the field shown in Figure 8 will only move about 6.4 ø of latitude in one doubling time. This change is sufficiently small as to suggest that the mode could be qualitatively identified with a blocking pattern. For other cases, larger growth rate increases the ratio still further. If, however, more restrictive lateral boundary conditions forced the mode to be slightly off peak growth rate, the ratio would be less than 1. The mode would then be most readily identified with the usual traveling planetary waves.

The dominant components of this mode are wave numbers n =-3 and n-- + 1 in the geopotential wave •p' and wave number n =-1 in the thermal wave •Pr'-This follows the pattern found in the stationary mode, but with the center shifted from n = 0 to -1. The zonal phase speeds of these components are -t-2.0, -6.0, and +6.0 m/s, respectively. In the classical problem, temperature components of baroclinically unstable modes travel at the same phase speed as the geopotential components, so it is worth pointing out that all temperature components progress in this mode; thus at wave numbers 1 and 2 we find geopotential components retro- gressing while temperature components progress. This is like the behavior described by Pratt and Wallace [1976] in an observational study of the zonal propagation characteristics of mid-latitude disturbances. Willson [1975-1 reports similar findings. Thus we have a mechanism by which both planetary

thermal waves and planetary geopotential waves with the correct propagation characteristics can grow through baroclinic instability.

(3) Other modes. A handful of other modes appear in the results and are sufficiently unstable that they might be significant. The three most important ones have minimum e folding times in this main case analysis of 10.8, 12.9, and 13.8 days, respectively, as compared to about 9 days for the two dominant modes. The first of these is a mirror image mode corresponding to the "B" mode described above. When the basic state forcing is simplified to just the controling wave number 2 elements, these two modes have the same growth rate but are mirror imaged in latitude and propagate in opposite latitudinal directions. The second consists chiefly of wave number 1 components and has its growth rate peak at a slightly larger meridional scale. Its growth rate curve appears in the inset of Figure 4. The third of these modes has a wave number 4 geopotential and a wave number 2 thermal component and propagates at somewhat larger phase speed than the other modes. All the modes are maintained by the same process as the "B" mode and conform to the propagation characteristics described above.

We describe these modes partly to indicate that a variety of planetary modes arise that would likely contribute to the low- frequency variance of the atmosphere. Although the growth rates of some of these are fairly low, it must be remembered that the response to the basic state forcings in this main case yielded rather conservative amplitudes for the basic state stationary waves. Since growth rate is roughly proportional to basic state wave amplitude, a modest increase in this would yield more impressive growth rates for these modes. In addition, these modes would often be nonnegligible in an initial value problem, particularly the mirror image of the "B" mode. The pair would tend to be excited simultaneously, resulting in a pattern that would not propagate in latitude.

4. MAINTE•qANCE OF PLANETARY MODES

4a. Energetics

Figures 9a and b show the energy conversions affecting the growing perturbation in our model for the two dominant modes. In both cases the main energy pathway can be summa- rized as

As-+ A'--} K'

where A s is available potential energy (APE) of forced stationary wave, A' is APE of growing mode, and K' is kinetic energy of growing mode. Both modes grow by baroclinic conversion of stationary wave APE to perturbation APE through the convergence of zonal heat transport.

In Figure 9c we anticipate the discussion in section 4c

0 ø 90 180 90 0 ø EAST WEST

LONGITUDE

Fig. 8. Geopotential height of 500 mbar for the "B" mode over one cycle in latitude. The latitude origin is arbitrary.

7208 NEELIN AND LIN' BARI)CLINIC GENERATION OF PLANETARY WAVES

As .4=0 ' ' _

Az'02• • As .4=4 '" _, (b)

Ks

As .4.2 .o.o K S

Fig. 9. Perturbation energy cycle' (a) "A" mode for the principal case, (b) "B" mode for the principal case, (c) "B" mode when the zonal thermal wind is 50% supercritical.

slightly by considering the results obtained from a variation of the basic state. The energy cycle shown here is for the "B" mode when the basic state thermal wind is at 1.5 times the

critical shear. This is included to demonstrate that the more

conventional baroclinic conversion of zonal APE (Az) to perturbation APE can also contribute to the growth of planetary wave modes when the shear is supercritical. However, this conversion from A z to A' is still of secondary importance compared to the A s to A' conversion, and it is less effective in the stationary mode than in the traveling ones. It is possible that in a model with better vertical resolution conversion of

A z might contribute slightly more to the growth rate of planetary modes. In our two-level model, planetary Rossby waves are far from the region of instability by conventional baroclinic growth from the zonal flow. In a vertically continuous model, planetary Green modes can draw energy from the zonal APE, although with low growth rate [Green, 1960].

We are now at a point where the characteristic form of the planetary wave Fourier spectra may be understood. The main available source of energy is the APE of the basic state stationary waves, since the zonal flow is not baroclinically unstable to perturbations of planetary scale for reasonable values of the vertical shear. When the energy cycle is examined at each wave number, it is found that in order for there to be a conversion of A s at wave number rn into ,4' at wave number n, C(Asm , An' ) a q/component must exist at either wave number n q-rn or n- rn with complex amplitude X,+m or X,-m, in addition to the q•r' component at wave number n with amplitude Y•:

C(Asm , An' ) oc [AmTrn*Xn_m + AmTrn*Xn+m]

Physically, the process is the advection of the wave number rn stationary wave temperature field by the wave number n _ rn perturbation mean flow field correlating with the wave number n perturbation temperature field. The characteristic form of the planetary modes is well adapted to this conversion, since there is a central thermal component at n flanked by two prominent geopotential components at n_ 2. This permits the tapping of APE from the predominant basic state wave at wave number 2.

4b. Planetary Mode Generation

To isolate the most important mechanisms involved in generating the planetary modes and to test the robustness of the

results obtained for the principal case, the instability analysis was carried out for a number of different basic states. In order

to vary parameters independently, some of the experiments were carried out by departing from the stationary equilibrium model and simply prescribing appropriate amplitudes for the basic state waves. The basic state was also varied within the

context of the stationary equilibrium model by changing the zonal mean and thermal wind, by increasing slightly the magnitude of the diabatic forcing, and by truncating different elements from the forcing. The details of these experiments are described in Neelin [1983], and we merely summarize the sa- lient results. The processes affecting planetary mode generation are dealt with in this section and the sensitivity of the results in the next.

(1) Role of basic state wave components. The growth of planetary modes is entirely supported by the temperature components Am T of the basic state waves. The geopotential components alone prove incapable of generating significant growth by barotropic instability, and they actually interfere with the baroclinic instability of the temperature waves, decreasing the growth rate about 10% in the principal case. Thus the geopotential waves create a threshold for the instability. For instance, when the diabatic heating is suppressed from the principal case, the topographic forcing still maintains a temperature wave of almost half the principal case amplitude, but because of the geopotential wave, no significant instability occurs. Conversely, an increase in diabatic heating produces an almost proportional increase in growth rate as a result of the increase in Am T.

The essential form of the principal case results can be reproduced by using only the dominant wave number 2 forcings in the basic state. The wave number 3 forcings have considerably smaller amplitude, and the wave number 1 forcings are less effective at generating instability.

(2) Role of direct interaction with topography. The direct interaction of the perturbation with topography plays a rea- sonably important role in producing the principal case planetary modes. When this interaction is artificially suppressed, the growth rate of the "B" mode is decreased slightly and its phase speed increases by about 70%. The "A" mode is more strongly affected, suffering a decrease in growth rate to about one third the principal case value.

The means by which the direct topographic interaction pro- motes planetary wave instability in our model is interesting because it has never, to our knowledge, been addressed in studies of atmospheric energetics. Holopainen [1970], for instance, evaluates the conversion of Kz to K s and K s to Kr because of the pressure gradient across topography (where Kr refers to transient eddy kinetic energy). He finds that these are small and concludes that the contribution of the mountains to

the atmospheric energy budget is insignificant. In our results the most important effect of the mountains is via the conversion of A' to K'. The topography induces an additional component of the vertical velocity field at 500 mbar. Evaluation of the omega equation [Holton, 1979, p. 186] with the mountains included in the lower boundary condition leads to an additional term whose Fourier components are given by

W,(topog) = -p2#2KL/(n2K 2 + L 2 + 1) M

E ??l(Sn-m- Yn-m)hm/H m= -I•

The conversion of perturbation APE to perturbation kinetic

NEELIN AND LIN.' BAROCLINIC GENERATION OF PLANETARY WAVES 7209

energy depends on the correlation of the vertical velocity field with the temperature field

C(A', K')= -fo/P: • Re{Yn*Wn)

where * denotes complex conjugation. Accordingly, the component of this conversion resulting from { Yn* Wn(topog)} can be attributed directly to the interaction of the perturbation with topography. Evaluation of this conversion for the "A" mode shows that the topographically induced contribution constitutes 67% of the total for the principal case. In the "B" mode this conversion is, again, over 60% of the total at peak growth rate. Thus the triggering effect of the mountains on the baroclinic conversion can be quite pronounced. Under certain circumstances, direct interaction with topography can produce a weak instability, even when no basic state stationary waves are present, inducing an A: to A' conversion. In our results, however, it is the cooperation of the topographic interaction in the baroclinic conversion of As to K' via A' that is important.

When the energy budget of the perturbation is decomposed by wave number, it may be seen that direct interaction with topography also plays a role in redistributing kinetic energy among different wave numbers in the perturbation. Although the sum of this interconversion is zero, it strengthens the linking between wave numbers in the perturbation spectrum.

(3) Results of an analytical model. In order to better un- derstand the results of the full model an analytical version was examined in Neelin [1983]. Basic state forcing at a single wave number m was considered, bearing in mind that forcing at rn - 2 alone gives a good approximation to the results of the full model. For the case where all the "off-diagonal" elements of the matrix are considered to be of order •, a small parameter, it was found that the only modes that can have significant growth rate for small rn can be described by the form {Xn+ m, Yn, Xn-m}, where Y, and at least one of X, are of order unity. This is the characteristic planetary mode form noted earlier for the principal case. The full model was used to verify that it holds for rn = 1 and rn = 3 as well. This two- or three-element

set contains the crucial dynamic process of APE conversion from the basic state wave to perturbation. The same form arises when the A m terms and diagonal en and an terms are of order unity and the other off-diagonal terms are small.

In either case the interaction of a {Yn, Xn+m} pair with the basic state wave is greatly enhanced when the corresponding Rossby wave frequencies coalesce, creating a triad resonance. For baroclinic instability of the basic state wave a necessary condition is that the meridional scale of the perturbation, l-x, be larger than or on the order of the radius of deformation. This condition is analogous to the short wave cutoff, which applies to the zonal scale of unstable modes in the classical zonally symmetric problem. When the triad resonance condition is met at a value of l that satisfies the necessary condition, it produces a peak in growth rate at this preferred meridional scale. For planetary modes centered at n- 0 or n = 1 the resonance condition can be met simultaneously by both pairs, so that all three perturbation elements participate in the resonance. For n = 2, only one pair can meet the resonance condition, and a {Y2, X,•} mode results, as described under "other modes" in the principal case. For a basic state wave of planetary scale, synoptic scale modes cannot meet both the resonance condition and the "cut-off" condition

simultaneously and thus have small growth rates, even for relatively large amplitudes of the basic state wave.

The resonance condition for the preferred meridional scale yields a fairly complicated dependence on U and rn. However, for realistic values of the zonal wind and planetary stationary waves this scale remains near the Principal case value of twice the radius of deformation. A very similar scale selection process is discussed by Pedlosky [1975a] for instability of oceanic baroclinic Rossby waves.

4c. Sensitivity of the Principal Case Results

In addition to the results described in the previous section, a number of other experiments were performed to test the sensitivity of planetary mode instability to variations of the basic state. Overall, we find that the principal case results hold qualitatively over the region of parameter space that is climatologically reasonable. The growth rates of the modes vary for different basic states, depending primarily on the amplitude of the basic state temperature waves, but the fundamental structure and phase speed of the modes are quite stable, except in one case. The "A" and "B" mode structures are prevalent because wave number 2 dominates the basic state for almost

all cases considered, combined with the constraints discussed in section 3b(3).

Variation of the zonal mean wind U provides a good exam- ple of this continuity in planetary mode structure. Increasing or decreasing U with fixed stationary wave amplitudes causes only a shift in the preferred meridional scale of the modes accompanied by small changes in growth rate. When stationary waves are dependent on U, according to the steady state model, changes in this parameter are more important. Increas- ing U from 3.0 x UTcri t to 4.0 x UTcri t reduces the Am T components and increases the A m components of the basic state slightly, resulting in roughly a 30% decrease in growth rate. Decreasing U to 2.5 x Urc,•it = 10.7 m/s has the reverse effect on the basic state, with the result that the "B" mode peak growth rate increases to 1/6 day-• and the "A" mode to 1/8 day-•. In both these cases the structure of the modes is essentially unaltered, although the "B" mode phase speed slows as its growth rate increases.

An exceptional case occurs when U is decreased to 2.0 x U refit = 8.6 m/s. The basic state thermal wave components

at wave numbers 1 and 2 increase to 91 m, as compared to 55 and 59 m in the principal case, while the geopotential components decrease by half. The resulting peak planetary mode growth rates are about 1/3 day-•; almost triple the principal case values. The form of the modes is also changed. Appar- ently the m = 1 linking between perturbation wave numbers is as important as the m = 2 linking for this case; all components from wave numbers 0 to 3 are of almost equal importance in the spectra. It is likely that the nonlinear effects of simultaneous interaction with two basic state wave numbers

have become important here. When only one of the wave numbers is included at a time, the growth rate increase over the principal case is not so dramatic. All the significantly unstable planetary modes in this problem are quasi-stationary in the sense we defined in section 3. Most have growth rates an order of magnitude larger than their angular frequency.

We hesitate to put much emphasis on these latter results because they are largely due to a quasi-resonant response of the basic state model. The steady state model has a resonance near U = UTeri t at which the temperature waves have a large amplitude response to the diabatic forcings, while the geopotential wave amplitude tends to zero. It is interesting that this resonance should affect our results while the more commonly


considered resonance of the geopotential wave does not [Tung and Lindzen, 1979]. However, the two-level steady state model response cannot be regarded as reliable with respect to these resonances because of poor vertical resolution and the effect of the lid [Kirkwood and Derome, 1977].

One of the most sensitive parameters in the conventional two-level model baroclinic instability problem is the zonal thermal wind. For the principal case we have used the critical value that lies at the threshold of instability when no basic state waves are present. When we increase the thermal wind above the critical value, significantly unstable synoptic scale waves appear among the normal modes. Their growth rate increases rapidly with the supercriticality, so when the shear is 50% supercritical, they are growing much more rapidly than the planetary modes. The synoptic scale modes are essentially the same as in the zonally symmetric problem and are modified very little by the presence of the basic state waves, even when the amplitude of these is quite large. The basic state waves have no meridional gradients and so do not act to increase the instability of the zonal flow anywhere. Appar- ently, the interaction of the unstable synoptic waves with the zonal gradients is quite weak. We can conclude that the preferred locations for cyclogenesis found by other investigators [Frederiksen, 1979] depend almost solely on the enhancement of the meridional temperature gradient by stationary waves with meridional dependence.

Although the synoptic scale modes grow faster than the planetary scale ones for these conditions, the planetary modes still exist. Upon examining the behavior of the planetary modes with respect to the zonal thermal wind, we find that they do grow faster when the zonal flow is less stable. Because the zonal APE is not so important to their maintenance, the increase in growth rate with supercriticality is less rapid than for synoptic modes. Conversely, when the thermal wind is decreased, or even set to zero, the growth rate and structure of these modes is scarcely affected.

Above critical shear, we thus have coexistence of two very different types of mode. The distinct separation in scale of the two types of modes--in both frequency and wave number-- implies that both can be important in an initial value problem, despite different growth rates. A transient pulse containing primarily planetary spatial components will excite planetary modes preferentially over the shorter scale modes. In the space of only a few e folding times before the disturbances reach maturity the larger growth rate of the synoptic waves will not always be sufficient to dominate the flow. The two types of modes have independent energy sources and thus do not com- pete for basic state energy. We suggest that the planetary modes will still be significant, even when conventional baroclinic instability yields faster-growing synoptic modes.

5. APPLICATIONS TO BLOCKING

So far in our discussion we have stressed applications of the forced wave instability process to generation of planetary waves, both stationary and traveling. We find certain of the planetary modes stationary or quasi-stationary, and so they lend themselves to being interpreted as growing blocking patterns. Thus the initial growth stage of the blocking phenomenon could be taken as a special case of planetary wave generation by forced wave instability.

Our stationary and quasi-stationary modes are composed primarily of wave numbers 1, 2, and 3. In this our results coincide with recent observational studies. Austin [1980]

found that the commencement of blocking periods is characterized by growth of stationary planetary waves to amplitudes much greater than normal, but with their normal phases. According to his study, zonal wave numbers 1, 2, and 3 account for the early stages of blocking, with interference of these components producing Atlantic, Pacific, or double blocking, depending on the relative amplitudes. A comprehen- sive study by Knox [1982] confirms that wave number 1, 2, and 3 components dominate the 500-mbar height field of blocking patterns with different relative amplitudes for different types of blocking. Concerning the phase of the waves, Knox found that at 60øN, wave number 2 components have their normal phase for double blocking, Pacific blocking, and "western European blocking," and approximately normal phase for "north-east Atlantic blocking," using his terminol- ogy. Wave numbers 1 and 3 have more variable phases, depending on the type of blocking. We have already pointed out that the stationary mode in our analysis (the "A" mode of section 3) is dominated by wave number 2 components with the zonal phase of the observed stationary waves. Therefore, the behavior of this mode parallels that of amplifying wave number 2 components in blocking.

The accord between the stationary mode and blocking components is most apparent for the case of double blocking. This category of blocking, in which blocks occur in both western European and Pacific sectors, has been cited by various investigators since Rex [1950]. Although not as common as regional blocks, instances of it occur with reasonable frequency. ½harney et al. [1981] found that four out of fourteen persist- ent blocking events for the period 1963-1977 were of this form. Because wave number 2 accounts for over 50% of the

stationary wave variance and has its normal phase [Knox, 1982], double blocking strongly resembles the normal stationary wave pattern, but with increased amplitude. Charney et al. present longitude plots of the latitudinal average from 50øN to 60øN for seven double-blocking situations. Not surprisingly, the stationary mode pattern shown in Figure 5 fits these well, since both resemble the normal pattern.

To better illustrate the resemblance of the stationary mode to double blocking, we set its amplitude to slightly over 200 m (since Knox gives the amplitude of the wave number 2 component to be 206 m at 60øN) and superimposed it on a constant zonal flow of 13 m/s. The resulting circulation is shown in Figure 10. The latitude band shown corresponds to one meridional wavelength of the mode. Two prominent blocking dipoles appear in the correct positions, separated by regions of zonal jets that have been enhanced by the stationary mode. It is worth noting that the apparent half wavelength of the blocking dipoles, if estimated by eye from the figure, would be only slightly over 20 ø of latitude, whereas the true half wavelength is 29 ø . The amplitude of the stationary mode relative to the zonal flow is also important to the appearance of the circulation. When the amplitude of the stationary mode is reduced to that of the normal stationary wave field, the dipole nature of the mode is not so evident, and the flow is more like the normal pattern.

The quasi-stationary mode (the "B" mode described in section 3) also has the qualitative characteristics of blocking but does not, by itself, resemble any observed blocking pattern. However, an initial transient forcing that is regional in extent will excite a set of modes simultaneously. In particular, the two dominant modes will grow in combination, interfering constructively in the region of excitation. This is in accord with the observations of Austin [-1980], mentioned earlier.


79•N

50*N

21*N

0 ø EAST 90 180 90 WEST 0 ø LONGITUDE

EURASIA N. AMERICA

Fig. 10. Stream function of 500 mbar for stationary mode plus zonal flow. The latitude band corresponds to one perturbation cycle. The latitude origin has been chosen to illustrate blocking. In geopotential units the contour interval is approximately 10 decameters.

To test this idea, we combine the two modes to see if they can produce regional blocking by interference. We have a lim- ited amount of freedom in combining them because the relative latitude origin can be varied through a full cycle. Figure 11 shows how Pacific blocking can be produced in this manner. Knox [1982] gives the amplitudes of wave numbers 1, 2, and 3 as 78, 121, and 102 m, respectively, for Pacific blocking, so we have set the amplitude of both modes to approximately 100 m. Destructive interference of wave number 1 components between the modes does reduce the wave number 1 amplitude somewhat in the total. The resulting flow in Figure 11 shows the correct strong zonal flow upstream of the block split by the blocking dipole over the eastern Pacific. Nonzonal circulation is small away from the blocking feature.

Combination of the two modes can also produce a regional block over Europe by shifting the quasi-stationary mode by 135 ø of meridional phase. The split in the jet occurs at about 10øW, which is reasonable, but this feature is less well localized, disrupting the flow over Asia as well. An important weak point of these normal modes as an explanation of blocking is that one can also produce a strong blocklike pattern over the western Pacific by shifting the stationary mode away from normal by 180 ø of meridional phase. Observations indicate that this area has, to the contrary, low frequency of blocking activity [Knox, 1982].

We have presented these results in such a manner as to suggest that the planetary modes of forced wave instability theory can potentially account for the growth stage of blocking. However, we must stress several reservations on this hypothesis. First, while we can roughly reproduce observed blocking patterns, we can also obtain patterns reminiscent of blocking, but with no observed counterpart. There is no el½-

ment in linear theory capable of distinguishing between these. Second, the amplitude of the waves during blocking is beyond the linear regime for which these results have been obtained. It is not obvious that the energy pathway found in the linear analysis can sustain their growth to large amplitude. A third deficiency precluding this theory from offering a complete de- scription of blocking is the lack of a mechanism to determine the conditions under which blocking will or will not occur.

A partial response can be given the second of these questions on the basis of general circulation model simulations of blocking. Kikuchi [19.69] and Chen and Shukla [1983] both obtain wave number 2 dominated blocking events during January simulations. The kinetic energy of the amplifying wave number 2 is supplied by conversion from available potential energy, which is the energy pathway found in our results. In both cases the land-sea heating contrast appears to play an important role. In the Chen and Shukla simulation a large-scale sea surface temperature anomaly that they intro- duced in the North Pacific was a critical factor. Kikuchi re-

marked that the baroclinic conversion at wave number 2 was

"inexplicable" in terms of the current baroclinic instability theory, suggesting that a new development in the theory was required. The forced wave instability process we present is consistent with these results. Naturally, the GCM blocking events are more complicated than can be accounted for by this single process. For instance, Chen and Shukla find that wave number 3 is also important to the block and is largely supported by conversion from the zonal kinetic energy. However, the GCM simulations do seem to justify the application of the linear theory to blocking as a partial explanation of some of the processes involved.

Concerning the third reservation, the linear theory would be

79•N

50*N

21eN

Oø EAST 90 180 90 WEST O* LONGITUDE

Fig. 11. As in Figure 10, for superposition of the stationary and quasi-stationary modes plus zonal flow.


more persuasive if it supplied insight into the conditions that were propitious or unfavorable to the onset of blocking. Fre- deriksen [1982] considered that planetary mode growth could be neglected when the zonal flow was unstable to fast-growing synoptic scale modes through the conventional baroclinic instability process. We have suggested that, as a result of separation of scales and the different energy sources for the two types of mode, this would not necessarily be the case. The planetary mode growth rate does, in fact, increase when the zonal flow becomes less stable. Furthermore, large-scale modes tend to grow to larger amplitude when finite amplitude effects are included [Gall, 1976]. We find that planetary mode growth occurs over a large range of basic state parameters. Were one to identify these modes only with blocking, the atmosphere should always be blocked. Instead, we propose that the planetary modes of forced wave instability theory should normally be identified with the observed transient planetary waves and components of the observed stationary planetary waves, both of which indeed dominate the atmospheric energy spectra at all times. The special case of blocking would arise when one or more of these modes happened to grow to much larger amplitude than normal.

Whether or not a growing planetary wave reaches blocking amplitude will be determined by the finite amplitude evolution of the system. It is conceivable that interaction with synoptic scale modes could act to suppress blocking, in which case Frederiksen's [1982] hypothesis would hold with only slight modification. A preliminary attempt to assess this possibility did not yield any evidence for it. Instead, we suggest that the controlling factor may lie in the nonlinear equilibration of the planetary mode with the basic state. We have pointed out certain analogies between our linear instability problem and one studied by Pedlosky [1975a]. Pedlosky [1975b] further examined the finite amplitude evolution of a triad consisting of a basic state marginally unstable Rossby wave and two perturbation components that correspond to a simplified version of our unstable modes. The maximum amplitude attained by the daughter waves was determined by nonlinear equilibration with the basic state wave. Although his findings do not apply directly to the system we are treating, a similar equilibration must occur. The question is, under what conditions would it occur at large amplitude. This amplitude would likely depend on how effectively the diabatic forcing supports the stationary wave APE against the growing dissi- pation by the planetary mode. The efficiency of this support and the details of the equilibration in turn determine the nonlinear behavior of both the forced stationary wave and planetary mode.

Although this equilibration is beyond the realm of linear theory, we can hypothesize, on the basis of growth rate, certain circumstances that might favor a block. The peak growth rate of the planetary modes is approximately proportional to the amplitude of the basic state thermal wave which in turn is roughly proportional to the diabatic heating forcing. Unus- ually large diabatic heating forcing should then be conducive to the initiation of a block, as indicated by the simulation of Chen and Shukla [1983]. Another factor that can increase the growth rate of planetary modes in our model is a decrease in the zonal mean wind. The increase occurs largely because the temperature wave in the basic state model increases in amplitude as the mean wind moves toward a resonant value. Be-

cause the basic state model is so simple, we consider these results to be reasonable only for values of the zonal wind that are not near the resonance. Although this conjectured factor is

rather tentative, it does have some correspondence with observation. White and Clark [1975] found a negative correlation between blocking and lower tropospheric wind for the years 1950-1970. Their results indicate that for seasons when the

700-mbar and surface winds are lower than normal there is a

greater probability of blocking. To ensure that the lower wind values were not caused by the blocking events, they repeated the calculations with the blocking periods excluded from the time average and obtained the same results.

Investigation of the nonlinear equilibration of these unstable planetary modes would provide a good test of the value of the proposed forced wave instability theory of blocking. If the planetary modes prove capable of reaching blocking amplitudes under these apparently favorable conditions, it would argue strongly for the usefulness of the linear theory.

6. CONCLUSIONS

We have taken a simple stationary equilibrium model of the northern hemisphere stationary waves and performed an instability analysis for the case of basic state forcings of long mcridional extent. The most unstable normal modes were

examined as an indication of the further evolution of the flow.

This main experiment was supplemented by a number of others aimed at determining the sensitivity of the response and the nature of the processes involved. We obtain results bearing on the related phenomena of generation of transient and stationary planetary waves and blocking.

First, we find that zonally asymmetric forcings can provide a source for planetary transient waves. A variety of unstable traveling planetary modes occur in our analysis, growing by baroclinic instability from the forced waves. When the zonal wind shear is sufficiently large, zonal available potential energy can provide a secondary energy source for these modes. These results concur with the earlier suggestions of Lin [1980b] and Sasamori and Youngblur [1981]. This mechanism for the generation of planetary waves provides a plausible explanation for the prevalence of these large scales in transient eddy statistics. A comparison with observed low-frequency power for the winter season suggests that the dominant components have similar meridional and zonal scales to those found in our results. Observational studies by both Youngblut and Sasamori [1980] and Lau [1979] confirm that transient planetary waves provide an energy sink for stationary wave available potential energy.

Second, one of the most interesting results of this study suggests that forcings of long meridional wavelength can contribute quite directly to the observed stationary wave pattern at shorter north-south scale. We find that our basic state gives rise, through baroclinic instability of the forced waves, to a stationary growing mode of the same form as the observed stationary wave pattern. Whereas in the stationary equilibrium model the meridional wavelength of the mountain and heating fields has to be suitably adjusted to obtain agreement with the observed distribution, the structure of the growing stationary mode is determined internally by the dynamics. Preferential growth rate selects a favored meridional scale of about twice the radius of deformation, which corresponds to a half wavelength of about 30 ø of latitude. This is the same scale as is necessary to obtain a good fit in the steady state model.

On this basis we postulate that a component of the time mean stationary wave pattern may be contributed by forcings of longer meridional scale. Repeated growth and decay of the stationary mode could yield a statistical contribution of the appropriate form to the time average. While the hypothesis


that transient waves supply a statistical component of the stationary waves is not new [Stone, 1977], the appearance of the stationary planetary mode in our results contributes a new variant on this idea. It is reasonable to suppose that, when all meridional scales are present in the forcing, both a directly forced component and a statistical component would coexist. The former would be a response to forcings of shorter meridional scale, conforming to the conventional steady state picture; the latter would be maintained by forcings of longer meridional scale through the instability process found here.

The inclusion of direct interaction of the perturbation with topography in our model is relevant to the maintenance of the stationary mode. The growth rate of this mode is strongly dependent on this interaction to trigger the baroclinic conversion that sustains it. Direct interaction with topography also affects certain of the traveling planetary modes, slowing their phase speed and increasing their growth rate slightly. In general the topographic effect assists in promoting wave number 1, 2, and 3 growth over that of intermediate and synoptic scales.

The third area to which our results may have an important application is the phenomenon of blocking. The growing planetary modes we find bear suggestive resemblance to blocking patterns. On the surface, therefore, our results appear to extend Frederiksen's [1982] conjecture that the onset of blocking can be explained in terms of linear instability of a zonally asymmetric basic state. By including direct interaction of the perturbation with topography, we obtained significant growth rates for planetary modes not seen by Fredericksen, providing a new variation on the hypothesis. The longer zonal scale and low phase speeds of these modes enhance their similarity to blocking as defined by Austin [1980]. The preferred meridional scale of our modes is consistent with the scale of blocking dipole high-low vortex pairs.

We do not feel, however, that we can make an unqualified identification of these planetary modes with blocking. The instability mechanism present here generates planetary waves almost irrespective of the basic state parameters. Were these planetary modes always to develop into blocking patterns, the atmosphere would be in a blocked state more often than not. We suggest, rather, that the growing planetary modes would usually appear as the normal transient planetary waves and components of the time average stationary waves. Blocking would occur as a special case when one or more of the modes achieved unusually large amplitude. Unfortunately, the linear theory does not seem to supply an obvious controlling factor that would determine the difference between normal generation of planetary waves and the creation of a blocking situ- ation. One hypothesis, advanced by Frederiksen [1982], is that blocking may be inhibited when the zonal flow becomes unstable to synoptic scale modes. As a possible alternative conjecture we propose that the regulating factor is found in the finite amplitude equilibration of the growing planetary mode with the forced wave that supports its growth in a manner analagous to that found by Pedlosky [1975b] for oceanic gyres. The excitation of one of the planetary modes found in linear theory would thus be a necessary condition to the onset of blocking, with the sufficient condition being determined by this hypothesized regulating mechanism. In most cases, we would expect that the planetary mode would equilibrate at relatively low amplitude, merely adding to the usual planetary wave variance. Our results suggest that one factor that should affect the propensity of the planetary modes to amplify into blocks is the magnitude of the land-sea diabatic heating. Low

zonal mean wind may also favor stronger than usual growth of planetary modes.

NOTATION

am, am r basic state stationary wave Fourier coefficients. f = fo + fly;Coriolis parameter on the beta plane. h height of topography.

H scale height. J(,) Jacobian with respect to x, y.

k = 1/(a cos (P0). Fundamental zonal wave number at latitude •00 (where a is radius of earth).

K = R/cp; ratio of gas constant to specific heat at constant pressure.

K, L, U, Ur, Am, Am r nondimensionalized k, l, u, ur, am, am r. I meridional wave number of perturbation.

m zonal wave number of basic state wave as multiple of k.

n zonal wave number of perturbation as multiple of k. P pressure.

Ap pressure difference between principal levels (500 mbar).

Q diabatic heating. R gas constant for air. t time.

T, 0 temperature, potential temperature. u, ur zonal mean and thermal wind. urcrit minimum critical shear value of u (=

V• = fxV(O/fo); horizontal geostrophic velocity. x, y horizontal beta-plane coordinates (east/north).

X,, Y• perturbation Fourier coefficients of •', •r'. AY half wavelength of basic state waves.

AY' half wavelength of perturbation modes. 2• = a/(kU); nondimensionalized frequency eigenvalue. /• inverse radius of deformation (/•2 = 2fo2/Ra•Ap). a complex frequency of perturbation.

a•, a• Re {a), Im {a); a• = growth rate. a: = - T/O c•O/c•p; static stability parameter. (Po central latitude of beta plane (45øN). • geopotential. • = «(•x q- •3); mean stream function.

•x, •3 geostrophic stream functions at 250 mbar and 750 mbar (• = fo- • •, i -- 1, 3).

•r = «(• - •3); thermal stream function. •', •r' time-dependent perturbation components of •, •r. •P, •Pr Basic state components of •, •r.

co vertical p velocity. co• co at bottom of model atmosphere.

Acknowlegments. This work was supported by a postgraduate scholarship (JDN) and research grant A4993 (CAL) from the Natural Sciences and Engineering Research Council of Canada and research grant 8380-1 (CAL) from Environment Canada.

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C. A. Lin, Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7.

J. D. Neelin, Geophysical Fluid Dynamics Laboratory, Princeton University, Princeton, NJ 08540.

(Received November 17, 1983; revised March 15, 1984;

accepted March 21, 1984.)

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