The importance of nonlinear wave processes in a quiescent winter stratosphere

QUARTERLY JOURNAL OF THE

ROYAL METEOROLOGICAL SOCIETY

Vol. 110 APRIL 1984 No. 464

Quart. J . R. Met. SOC. (1984), 110, pp. 289-301 551.510.532:551.513.1

The importance of nonlinear wave processes in a quiescent winter stratosphere

By JOHN AUSTIN and T. N. PALMER Meteorological O f i c e , Bracknell

(Received 20 April 1983; revised 1 November 1983)

SUMMARY Linear wave modelling about a zonally symmetric state is often assumed to be able to simulate accurately

time-averaged planetary wave geopotential heights in the stratosphere during quiescent winter periods. We test this linear assumption by using a time-dependent three-dimensional primitive equation model of the stratosphere and mesosphere to simulate the quiescent winter stratospheric circulation during December 1980. The results from this simulation, together with two other idealized integrations, indicate that nonlinear wave interactions were crucial in accounting for the observed monthly mean wavenumber-1 planetary wave amplitude, particularly in the upper stratosphere. Our results suggest that, in general, such interactions cannot be ignored for quantitatively accurate simulations of the middle atmosphere.

1. INTRODUCTION

It is now well established (Hsu 1980; Lordi et al. 1980; Palmer and Hsu 1983) that in order to model wavenumber-1 amplitudes accurately in the stratosphere during sudden warming conditions, nonlinear wave interactions must be taken into account. On the other hand, it is often assumed (Holton 1975) that stationary linear wave modelling about a zonally symmetric state, as pioneered by Matsuno (1970), is sufficient to simulate time-averaged planetary wave geopotential heights in the stratosphere during quiescent winter periods, i.e. periods when the wave amplitudes are slowly varying.

One such winter period was observed during December 1980, using data from the Stratospheric Sounding Unit on board NOAA-6. In general, without the convergence of Eliassen-Palm fluxes, the upper stratosphere tends towards a cold thermal equilibrium. Wave transients generate such convergence. The quiescence of the period, then, can be inferred from Labitzke and Goretzki’s (1982) remark that “the midwinter period Decem- ber 1980-January 1981 was the coldest winter in the northern hemisphere since 1955/ 56”.

In view of this it might be imagined that linear planetary wave theory ought to be able to model the observed monthly mean wave amplitudes for December 1980. However, when the stationary linear wave model of Austin (1982) was used to verify this, the model considerably underestimated observed wavenumber-1 amplitudes in the upper stratosphere. Austin (1983) has shown that the wave response in linear models may be very sensitive to the amount of wave transience present. However, his results have suggested that the inclusion of such wave transience into the model formulation would result in even smaller amplitudes than were obtained by neglecting the transients.

These results prompted us to consider whether nonlinear effects may have been important in accounting for the December 1980 monthly mean amplitudes. In order to test this, we have conducted a simulation and a series of idealized experiments using a

289

290 J. AUSTIN and T. N. PALMER

fully nonlinear time-dependent primitive equation model of the stratosphere and mesosphere. The model is an extended version of that used by Butchart et al. (1982).

When the model is initialized on 9 December and forced at its lower boundary with real data we find approximate agreement with the appropriate observations. In the first of two idealized experiments, all the wavenumber-1 forcing at the lower boundary is removed from 15 December onwards and we find that the wavenumber-1 response in the upper stratosphere is very similar to the full simulation. In the second idealized experiment the initial and boundary conditions are identical to those of the full simulation, and the effect of wave-wave interactions has been removed from the model. In this integration we find that the wavenumber-1 amplitude is underestimated by about a factor of two in the upper stratosphere. Indeed the wavenumber-1 response is similar to that predicted by Austin's linear stationary wave model.

These results demonstrate convincingly that wave-wave interactions in which wave 1 was generated in situ from higher wavenumbers were indeed crucial in accounting for wavenumber-1 amplitudes during this period. In view of these results we consider whether such nonlinear effects may play a more ubiquitous role in determining monthly mean planetary wave amplitudes than is conventionally thought.

2. OBSERVATIONS In this section we present some observations of the troposphere and stratosphere

during December 1980. Up to 100mb the data come from daily National Meteorological Centre (NMC) analyses; above 100mb we use radiance data from the Stratospheric Sounding Unit (Miller et al. 1980) on NOAA-6. Thicknesses for stratospheric layers up to 1 mb were retrieved from these radiances and converted to daily height fields using the NMC 100mb values. These daily height fields were averaged to obtain a monthly mean dataset.

Geostrophic mean zonal winds for the monthly mean fields are illustrated in Fig. 1. The polar night and subtropical jets are clearly shown, the former having a maximum

Latitude rN) Figure 1. Mean zonal wind (m s-') for December 1980.

of over 90ms-I at about 50"N. Monthly mean geopotential height amplitudes for wavenumbers 1-3 are presented in Fig. 2. The amplitude of wavenumber 1 has two maxima with a distinct amplitude minimum, near the tropopause, between latitudes 30 and 70 ON. This pattern of wavenumber-1 amplitude appears rather anomalous when

NONLINEAR WAVE PROCESSES 29 1

80 60 40 20 Latitude ("N) (b)

1

I n 10 E

E

E n. 100

I

'A #n

850 80 60 40 20

(C) Latitude rN) 1

- 10 n E

?!

E 0. 100

- 'A

850 20 40 80 60

Latitude rN)

Figure 2. Wave amplitudes (m) of the monthly mean geopotential height field. (a) Wavenumber 1; (b) wavenumber 2; (c) wavenumber 3.

compared with other December means in the years when SSU data are available, as recently documented by Geller et al. (1983). In the troposphere, wavenumber-1 amplitudes are weaker than normal in high latitudes having less than half their normal value in the high latitude upper troposphere (see Geller et al., Fig. 3). North of 50"N the wavenumber-2 amplitudes are continuously increasing from the troposphere to the stratosphere with amplitudes a little smaller than wavenumber-1 amplitudes in the upper stratosphere. These wavenumber-2 amplitudes are a little larger than average in the troposphere, especially in mid to low latitudes, and considerably larger than average in


1

s 10 E e e

I

v) .l

100 n

Latitude rN)

1

3 10 E ?

e a. loo

- v)

80 60 40 20 850

Latitude rN)

Figure 3. Eliassen-Palm flux divergence (108kgs.') and integral curves from the monthly mean height field. (a) Wavenumber 1; (b) wavenumber 2.

the upper stratosphere (see Geller et al., Fig. 4). Wavenumber-3 amplitudes are, in general, considerably smaller than those of either wavenumbers 1 or 2 in the stratosphere and for December 1980 had about average strength (see Geller et al., Fig. 5 ) .

Figure 3 shows integral curves (Palmer 1982) of the EP flux F and its divergence, V . F, for wavenumbers 1 and 2 of the monthly mean fields. Details of this diagnostic have been presented elsewhere (Edmon et al. 1980; Palmer 1981) and will not be repeated here. In Fig. 3 it can be seen that the largest wavenumber-1 flux convergences are roughly coincident with the amplitude maximum in the troposphere. The integral curves associated with these fluxes (shown dashed) are directed equatorward and contained within the troposphere. No net wavenumber-1 activity appears to be propagating up into the stratosphere, and there is a flux divergence in the mid-latitude stratosphere. As Fig. 3 shows, wavenumber-2 fluxes are directed polewards in the troposphere with a split in the integral curves in the lower stratosphere with some of the curves directed polewards and downwards and some directed upwards into the stratosphere where they curve back towards the equator. Compared with the normal direction of the stationary EP fluxes illustrated in Fig. 10 of Geller et al., the direction of the fluxes for December 1980 is anomalous particularly in so far as there is no indication of wavenumber-1 activity propagating into the stratosphere from the troposphere.

NONLINEAR WAVE PROCESSES 293

The monotonicity of amplitude with wavenumber in the stratosphere is conventionally explained with the aid of linear stationary wave theory (see, for example, Holton 1975) which predicts that the larger the zonal wavenumber the less the wave can propagate in the vertical. Such an explanation does not readily explain the observations discussed above; in particular the wavenumber-1 EP cross-section. Perhaps the most natural explanation for this apparent disparity is to invoke transient effects. For example in the absence of dissipation, linear wave theory would predict that if wavenumber-1 activity had been decaying in the stratosphere, the corresponding EP fluxes would have been divergent. However, although such a decay was noted for the first half of the month, this was not the case for the second half. This can be seen from Fig. 4 which shows that

O o 0 c b / j I

2001 \! n

Date December 1980 Figure 4. Observed wavenumber-1 amplitudes and forcing amplitude (m) for experiment 1 at 67.5 "N.

(a) Observed 100mb; (b) observed Imb; (c) wavenumber-I forcing for experiment 1.

the wavenumber-1 amplitudes, at 1 mb and 67.5 ON, were approximately the same at the end of the month as at the beginning of the month after reaching a minimum on 16 December. (This latitude was chosen since it corresponds to the latitude of maximum total wavenumber-1 amplitude at 100mb in Fig. 6(a).) Hence we were led to consider other explanations of the wavenumber-1 observations.

It is of interest to note that there was a period of strong growth in wavenumber 2 in the lower stratosphere around the middle of the month as shown for 67.5"N and 100 mb in Fig. 5 . Results of our model integrations, described in section 4, suggest that this may have led (through nonlinear effects) to the large increase in wavenumber 1 in the upper stratosphere during the last few days of the month (see Fig. 4).

3. DESCRIPTION OF THE MODEL INTEGRATIONS

The model is an extended version of that used by Butchart et al. (1982) and has 4th-order finite differencing (Kalnay-Rivas 1976) in the advection and pressure gradient terms, which has been shown to improve certain model deficiencies near the polar cap, as discussed by Butchart et al. (S. A. Clough, personal communication). In addition, the model has an improved radiation scheme with three explicit components: parametrized


1, 11 13 15 17 19 21 23 25 27 29 1 I 1 1 1 I l l I J

Data D.camb.r lPB0

Figure 5. Observed wavenumber-2 amplitude (m) at 67.5%. 100mb.

solar heating, parametrized 9.6 pm band 0 3 cooling and a 15 pm band COZ cooling term calculated with a Curtis matrix. The solar heating rates are calculated using the parametrization of Schoeberl and Strobe1 (1978) using a constant zonally meaned ozone concentration with the solar declination fixed at the solstice value. The 9.6 pm 0 3 cooling term was obtained from the parametrization of Harwood and Pyle (1975).

The model has a horizontal resolution of 5 O of latitude between 87.5 ON and 87.5 "S and has 16 points around each latitude circle. The vertical points are equally spaced in log(pressure) with 33 points between 0.01 and 100mb where the model is forced with observed (NMC) or idealized geopotential heights. Initial conditions were taken from observed data below 1 mb and extrapolated above using the same procedure as outlined in Butchart et al.

A 3-week integration is as long as one could realistically expect the model to reproduce the main features of the observations, possibly because of spurious resonances (McIntyre 1982). As we were particularly interested in the second half of December, we decided not to initialize the integration from the beginning of the month. Also the planetary wavenumber-1 amplitude became very small (less than 50 m) in high latitudes at 100 mb on 15 December, as shown for 67.5 ON in Fig. 4(a). Because this was a factor which could be used to advantage in one of the idealized runs (described below) the December 1980 simulations were based around this data. Further, since the model typically takes several days to adjust to the geostrophic initialization, a 'running-in' period of five days was added. Hence, the first two integrations were started from the observed geostrophic fields on 9 December and forced with observed (NMC) 100mb geopotential for the five-day running-in period up to 14 December 1980. In the full simulation, the integration was continued while still forcing with real data until 29 December. This date was chosen because data for 30 and 31 December were not available.

In experiment 1 the integration was continued from 14 December but with wavenumber 1 filtered from the lower boundary forcing data from 15 December onwards (Fig. 4(c)). For a stable mean flow, when wavenumber 1 is eliminated from the lower boundary it can only be sustained in the model interior by nonlinear interactions. Consequently if nonlinear effects dominate, both experiments should give broadly similar results. Note that the existence of a very low amplitude in wavenumber 1 at the lower boundary on 15 December ensures minimal disturbance to the model when the integration is continued. If wavenumber 1 were removed from the lower boundary from 9 December


onwards, large gradients in wavenumber 1 near the lower boundary, due to the existence of wavenumber 1 in the initial state, might produce spurious results.

Experiment 2 was designed to provide further evidence of the importance of nonlinear effects. In this experiment the model was initialized with 9 December data which had been Fourier filtered leaving wavenumber 1 and the mean flow only. The lower boundary condition for the model was the NMC 100mb field as used in the simulation. The model was then integrated up to 29 December applying the same Fourier filter at each timestep to remove all waves shorter than wavenumber 1, thus eliminating the possibility of generating wavenumber 1 by wave-wave interaction. This is similar to the model used by Holton and Wehrbein (1980).

The results from these experiments are described in the next section.

4. RESULTS OF THE MODEL INTEGRATIONS

Figures 6(b)-(d) and 7(b)-(c) show the wavenumber 1 and 2 response for the full model simulation and idealized experiments. The diagrams illustrate the daily values of wave amplitude averaged over the period of the experiments, 9-29 December, which includes the running-in period. For comparison, Figs. 6(a) and 7(a) show the corresponding observed fields.

Figure 6(b), for the full simulation, compares well with observations in the upper stratosphere. The peak wavenumber-1 amplitude at 1 mb is 600m, within 10% of the observed value, and at about the correct latitude. In the mid stratosphere the simulation has underestimated the observed amplitudes. At 10 mb, for example, the simulated peak wavenumber-1 amplitude is about 250m as compared with the observed value of 375 m.

Ibl

Figure 6. Wavenumber-1 amplitudes (m) averaged for the period 9-29 December 1980. (a) Observations; (b) model simulation; (c) experiment 1; (d) experiment 2.


Figure 6(c), for experiment 1, shows values remarkably similar to the full simulation. The overall wavenumber-1 maximum in the mesosphere is about 900m for experiment 1, as compared with 800m in the full simulation. At l m b the peak amplitude is about 675m and at 10mb the peak amplitude is about 300m. In fact these values compared quite well with the observations and in some respects are better than those of the full simulation. Figure 6(d), for experiment 2 (the quasi-linear simulation) shows amplitudes which, in the mesosphere and upper stratosphere, are considerably smaller than those for the simulation. For example, at 1 rnb, the peak amplitude is 350m, only 60% of the corresponding value for the full simulation.

These results demonstrate quite clearly that nonlinear effects are important, especially in the upper stratosphere. The quasi-linear simulation, experiment 2, is clearly

Latitude

(bl

Latituda

I .

4) 75 60 411 Y) I S

Latitude

Figure 7. Wavenumber-2 amplitudes (m) averaged for the period 9-29 December 1980. (a) Observations; (b) model simulation; (c) experiment 1 .


0

deficient; conversely above 10 mb the experiment forced mainly by wavenumber 2 at 100 mb produces very similar amplitudes to the full simulation.

The wavenumber-2 response for the full simulation is shown in Fig. 7(b). The amplitudes are somewhat larger than observations (Fig. 7(a)), as can be seen at l m b where the peak amplitude is 875 m compared with the observed 650 m. However, the model correctly predicts that in the upper stratosphere the wavenumber-2 response should be considerably larger than normal (cf. Geller et al. 1983, Fig. 4). At lOmb the peak amplitude is 400 m compared with the observed 300 m although the local maximum near this level is not reproduced by the model. Figure 7(c), for experiment 1, again shows very similar results to the full simulation. By construction the wavenumber-2 response for experiment 2 is identically zero.

It is interesting to note that in the upper stratosphere the wavenumber-1 amplitudes in experiment 1 are larger than in the simulation. Figure 8, showing the phase of wavenumber 1 at 67.5 O N for the simulation and idealized experiments, suggests a reason for this. Notice that the phases of wavenumber 1 for the simulation and experiment 1 follow one another closely in time, and both diverge from experiment 2 after December 19. One might therefore hypothesize that there is some phase cancellation, in the simulation, between the strong nonlinear contribution shown in experiment 1, and the weaker quasi-linear contribution shown in experiment 2. Figure 8 shows that although the day-to-day simulation of the phase of wavenumber 1 was not very good, the time- averaged phase was about correct.

\ \ \ b i.0

1 1 1 1 l l l h. I

Y 0 w

Y

Figure 8. Phase of wavenumber 1 at 67.5 ON, 1 mb. (a) Simulation; (b) experiment 1; (c) experiment 2; (d) observations.

Figure 9 shows a time plot of wavenumber-1 amplitude for the simulation and two experiments. Curve (a) shows that the simulation captured the broad features of the observations (curve (d)), particularly the large rise in amplitude towards the end of the

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J. AUSTIN and T. N. PALMER

/ b / I I

I 120ot

Date December 1980

Figure 9. Model wavenumber-1 amplitudes (m) at 1 mb 67.5 ON. (a) Simulation; (b) experiment 1; (c) experiment 2; (d) observations.

period. Curve (b), for experiment 1, exaggerated this rise possibly due to reasons already discussed. The quasi-linear experiment, shown by curve (c), shows an initial decrease from 900 m with the amplitude maintaining a fairly steady value of about 250 m over the last 10 days of the experiment. As shown in Fig. 5 , the observed wavenumber-2 amplitude at 100mb starts to increase on 14 December and peaks on 22 December and would be expected to take some 5-10 days to propagate to the upper stratosphere and mesosphere. It therefore seems likely that the increase in wavenumber 1 towards the end of the period can be associated with the nonlinear interaction of existing wave 1 with the main pulse of wavenumber-2 forcing at 100mb. By definition, this association could only occur through nonlinear wave interactions, and is therefore absent in curve (c) of Fig. 9.

5. CONCLUSIONS AND DISCUSSION

In this paper we have investigated the dynamical behaviour of a quiescent winter stratosphere observed in December 1980, using an extended version of the primitive equation model of Butchart et al. (1982). The model was run for the period 9-29 December while forcing with the observed 100mb geopotential height field. As noted in section 4 reasonable agreement with observation was achieved for wavenumber 1 though wavenumber-2 amplitudes were in general too large.

In experiment 1 the model was run again but with wavenumber 1 Fourier filtered from the 100 mb forcing field for 15 December onwards. In experiment 2 the model was run with the observed 100mb forcing field and with a Fourier filter applied at each timestep to eliminate all waves shorter than wavenumber 1. The latter experiment was thus equivalent to performing a simulation with a model which does not permit wave- wave interaction but which has wave-mean-flow interaction (e.g. Holton and Wehrbein 1980).

The simulation and experiment 1 produced very similar wavenumber-1 amplitudes in the mid and upper stratosphere and mesosphere despite the fact that from 15 December


these amplitudes were absent from the 100 mb forcing field of experiment 1. In experiment 2, by contrast, the wavenumber-1 amplitude decreased by a factor of approximately 3-4 in the first 10 days and remained approximately steady until the end of the integration. In comparison the observed wavenumber-1 amplitude initially decreased, and then increased from about the middle of the month so that by the end of the month the amplitude had recovered to its original value. This suggests that although linear theory may be adequate to explain the wavenumber-1 behaviour for the first half of the month nonlinear effects dominate for the second half of the month when the wavenumber-2 pulse (see Fig. 5) from the lower boundary reached the upper stratosphere.

Two physical mechanisms for wave-wave interactions have recently been described in the literature. During wave-breaking events (McIntyre and Palmer 1983), which one would expect from nonlinear critical layer theory to be occurring continuously around the zero wind line in the subtropics (see Fig. l), there would be an irreversible cascade of wave enstrophy to small scales, and the interaction coefficients between the higher zonal harmonics would be active. Towards the end of December 1980 wave breaking may have been occurring in the mid-latitude upper stratosphere, where, from Fig. 9, the waves appear to be most transient.

A second mechanism for wave-wave interactions is discussed in Palmer and Hsu (1983) and is based on the notion of wave activity propagating on a zonally asymmetric state. From a zonal average point of view this could only be described in terms of wave-wave interactions. A useful diagnostic scheme for studying the importance of such a mechanism remains to be developed, and at present we can only speculate that it was likely to be a significant factor in accounting for the observations. It is unlikely that a detailed diagnosis of the zonal harmonic interaction coefficients would elucidate the physical mechanism at play, and for this reason we have not performed such an analysis here. However, although wavenumber-3 geopotential amplitudes were observed to be considerably smaller than those of wavenumbers 1 or 2 (Fig. 2(c)), it cannot be inferred that interactions involving wavenumbers 2 and 3 were unimportant in generating wavenumber 1 (Smith 1983).

We have noted that the phases of the wav&umber-1 response in experiments 1 and 2 tended to be in partial cancellation. It is therefore possible that the minimum in the observed stationary wavenumber-1 response in the lower stratosphere (Fig. 2) can be thought of as arising from some cancellation of the weak tropospheric wavenumber-1 forcing and the component of stratospheric wavenumber-1 amplitude generated in situ by nonlinear interactions.

The month of December 1980 had unusually weak high latitude tropospheric stationary wavenumber-1 forcing. This situation undoubtedly helped us in obtaining such a clear signal from the nonlinear wave interactions. On the other hand it is of interest to assess the importance of such interactions for months when the tropospheric wavenumber-1 forcing was stronger compared with wavenumber 2. Under such circum- stances it is possible (from the above results) that the effect of nonlinear wave interactions is partially to damp the linear wavenumber-1 response in the lower and mid stratosphere though this would require the same phase relations between wavenumbers 1 and 2 as for December 1980. A damping effect could be incorporated into a linear model by means of an increased Rayleigh friction coefficient. However, a parametrization of nonlinearity by a friction coefficient would certainly not have the correct wavenumber dependence. We note in this respect that the value of the Rayleigh friction coefficient used by Matsuno (1970) was unrealistically large in the stratosphere (Crane et al. 1980). With this large friction coefficient, Matsuno found he was able to simulate accurately

300 J. AUSTIN and T. N. PALh4ER

wavenumber-1 disturbances, though he underestimated the wavenumber-2 response. Hence Matsuno’s integrations themselves do not rule out the possibility of nonlinear wave interactions being quantitatively important, although an increased friction coefficient might also partially represent transient effects which were not generated by nonlinearity, as suggested by Austin (1983). In general, however, since we have found nonlinear effects to be important during a period when wave amplitudes were relatively small, one might expect these effects to be significant during periods when wave amplitudes are larger.

ACKNOWLEDGMENTS

We would like to thank Drs G. Vaughan and J. Haigh for supplying a Curtis matrix for adapting to our model grid. We would also like to thank Drs M. E. McIntyre, J. R. Holton, R. A. Plumb and S. A. Clough for their comments on the material. Dr S. A. Clough also designed the integration scheme of the model. Finally, we would like to thank Mrs B. C. Kingston for supplying programming assistance.

Austin, J .

Butchart, N., Clough, S. A., Palmer, T. N. and Trevelyan, P. J.

Crane, A. J., Haigh, J. D., Pyle, J. A. and Rogers, C. F.

Edmon, H. J., Hoskins, B. J. and McIntyre, M. E.

Gelman, M. F. Geller, M. A., Mao-Fu, W. and

Harwood, R. S. and Pyle, J. A.

Holton, J. R.

Holton, J. R. and Wehrbein, W. M.

HSU, C-P. F.

Kalnay-Rivas, E.

Labitzke, K. and Goretzki, B.

Lordi, N. J., Kasahara, A. and Kao, S . K.

McIntyre, M. E.

McIntyre, M. E. and Palmer, T. N.

Matsuno, T.

1982

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1970

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importance of travelling wave components. Quart. J . R. Met. SOC., 108, 763-778.

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Mean meridional circulations of the stratosphere and mesosphere. Pageoph., 111, 307-328.

Eliassen-Palm cross-sections for the troposphere. J . Atmos. o Sci., 37,260LX2616. (see also corrigendum J . Amos. Sci.,

38, 1115 especially second last item). Troposphere-stratosphere (surface - 55 km) monthly winter

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Air parcel motions during a numerically simulated sudden stratospheric warming. ibid, 37, 2768-2792.

High latitude truncation errors of box type primitive equation models. Mon. Weath. Reu., 104, 1066-1069.

Middle Atmosphere Program Handbook for MAP, Vol. 5 , p. 26.

Numerical simulation of stratospheric sudden warmings with a primitive equation spectral model. J . Amos. Sci., 37, 2746-2767.

How well do we understand the dynamics of stratospheric warmings? J . Met. SOC. Japan, 60,3745

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Miller, D. E., Brownscombe, J. L., Carruthers, G. P., Pick, D. R. and Stewart, K. H.

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Operational temperature sounding of the stratosphere. Phil.

Aspects of a stratospheric sudden warming studied from a transformed Eulerian mean viewpoint. J. Geophys. Res., 86, 967!9-9687.

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Stratospheric sudden coolings and the role of nonlinear wave interactions in preconditioning the circumpolar flow. ibid, 40, 909-928.

The zonally averaged circulation of the middle atmosphere. ibid, 35, 577-591.

Observations of wave-wave interactions in the stratosphere. ibid., 40, 248424%.

Trans. R. SOC. Lond., A2%, 65-71.

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The importance of nonlinear wave processes in a quiescent winter stratosphere