Chapter I11 ESTIMATION OF EQUATORIAL WAVE MOMENTUM FLUXES USING MST RADAR MEASURED WINDS

Transcription

1 Chapter I11 ESTIMATION OF EQUATORIAL WAVE MOMENTUM FLUXES USING MST RADAR MEASURED WINDS 3.1. Introduction The dynamics of equatorial lower stratosphere is dominated by the well known QBO of the zonal wind. The basic mechanism responsible for the QBO is the wave-mean flow interaction wherein upward propagating equatorial wave, i.e., Kelvin and RG waves. inrer.act with the stratospheric mean flow and deposit their momentum producing the alternating westerly and easterly regimes respectively. An important parameter used for the simulation of the QBO is the prescription of - meridionally averaged vertical flux of horizontal momentum, u'w' (u' and w' are perturbations in zonal and vertical velocity) of the Kelvin and RG waves at the lower boundary (at a height of -17 km) of the model. Holton and Lindzen (1972) used a value of 4x 10.' mlr;" and -4~10.~ m2s" for the Kelvin and RG waves, respectively. However, the QBO is sensitive to any change in the wave flux. Plumb (1977) showed that the QBO period decreases with increase in wave momentum fluxes at the lower boundary. Dunkerron (1981) in his numerical simulation of the QBO found that, when the wave flux due to RG wave was reduced by a factor of 2, the QBO period increased by 22%. The reduction in RG wave flux increased the length of the westerly phase and reduced that of the easterly phase at 20 km and vice versa at 30 krn, resulting in an increase in the QBO period at both heights. Similarly the changes in Kelvin wave flux also could alter the lengths of the westerly and easterly phases of the QBO. Recently. Geller et al. (1997) have also done a similar study and shown that the QBO period changed from -25 months to -45 months when the Kelvin and RG wave momentum fluxes were changed from +8~10-~ m2s" to mzs-', the QBO period of '33 months corresponds to a flux value of k4~10"m~s~~. These investiga1:ions show the importance of an observation study of the momentum fluxes of equatorial waves, which are driving the QBO oscillations in

2 the equatorial lower stratospheric zonal winds. In this chapter a method to compute the momentum tluxes of equatorial waves from the radar winds will be presented. The method is developed using MST radar measured winds Estimation of Equatorial Wave Momentum Fluxes Using Radiosonde Data In earlier studies of the equatorial wave momentum fluxes radiosonde zonal wind and temperature data were used for their calculations (Maruyama, 1968; Wallace and Kousky ). In this method, the vertical flux of horizontal (zonal) -~ momentum u'w' is estimated using the zonal wind perturbation (u? obtained directly from the radiosonde zonal wind data and vertical wind perturbation (w3 obtained indirectly from the radiosonde temperature data. In these studies it is assumed that the temperature perturbation is caused by wave-associated adiabatic vertical motion and w' is derived from temperature perturbation using the first law of thermodynamics. These early studies used average amplitudes of zonal wind and temperature perturbations for deriv~ng the momentum flux. This method when - applied to radiosonde data for Canton Island (2.85, 171.7"W) gave u'w' to be -4x 10" m2s ' and - 1 x 10" m's ' for Kelvln and RG waves, respectively (Wallace and Kousky, 1968b). However, the direct vertical momentum fluxes due to both waves are positive, implying upward transport of westerly momentum for both. But in the case of RG waves indirect forcing due to the meridional transport of eddy heat flux produces a large easterly momentum, making the net forcing easterly. For Kelvin waves, this contribution will be zero since the wind perturbation in the meridional direction is zero for all these waves. The flux values of?4x1o3 m2s-2 for Kelvin and RG waves were used first by Holton and Lindzen (1972) in their QBO model and these were based on the estimated fluxes for Canton Island by Wallace and Kousky (1968b). Recently, Saro and Dunkerfon (1997) have estimated the momentum fluxes associated with the equatorial waves from rawinsonde data at Singapore. Sato and -- Dunkerfon (1997) calculated u'w' of the equatorial waves based on quadrature specua (Q,) of temperature (T1 and zonal wind (u).

3 The momentum flux is calculated from quadrature spectra Q, by the following equation. where w, and 0, are lower and upper limits of the frequency range of equatorial waves, g is the acceleration due to gravity,?is the mean background temperature - and N is the Brunt Vaisala frequency. They obtained u'w' as 2x10" - 9x10-3 mzs" for Kelvin waves (5-20 days) and 2x10.' - 6~10.~ rn'~.~ for RG waves. They also estimated z?from the cospectra C,, of T and u fluctuations where w, and wz are lower and upper limits of the frequency range of equatorial waves, g is the acceleration due to gravity, Tis the mean background temperature, N is the Brunt Vaisala frequency and U. strength of wind shear. From the cospectra - estimates of u'w' they obtained a value of 3x10" - 50x10" m2s-2 for Kelvin waves and 7x10" - 30x10' m's ' for RG waves Momentum Flux Calculations Using Radar Measured Winds With the advent of MST radars, it has become possible to measure directly the 3-dimensional winds in the troposphere and lower stratosphere. Thus one can obtain the zonal, meridional and vertical components of the winds, which can be, - made use of in deriving the rnomentum flux u'w' of the equatorial waves. Vincent and Reid (1983) devised and applied a technique using a dual-beam-doppler Medium Frequency (MF) raiiar for gravity wave momentum flux studies in the mesosphere. In the symmelric beam method of Vincent and Reid (1983), the momentum flux in a vertical plane is obtained using two radial radar beams in that plane, pointing at symmetric zenith angles, + 6 and r.; u'w' = r+~ 2 sin 28

4 where u'and w'represent ithe horizontal and vertical perturbation velocities and r +~ and r.0 represent radial perturbation velocities in the two beams. For instantaneous measurements, this would require the horizontal scale of the waves to be far greater than the spatial separation of the beams at the height of interest. However, for an averaging time much longer than the observed wave periods, the velocities recorded by each beam are assumed to be statistically similar; the momentum flux is given by the variances in the two beams. This method has been applied to measure the momentum flux in both the lower atmosphere and the mesosphere. Their technique was further utilised and refined by Fritts and Vincent (1987). Reid and Vincent (1987) and Murphy and Vincent (1993). MST radars have been used to obtain estimates of mesospheric momentum flux using this method. The Poker Flat radar was used for momentum flux measurements by Fritts and Yuan (1989) and Wang and Fritts (1991). the SOUSY MST radar by Reid et al. (1988) and by Rusfer and Reid (1990), and the MU radar in Japan by Tsuda et al. (1990). Another approach (e.g, Fukao ef al., 1988) uses the product of the perturbation vertical wind component measured directly with the vertical radar beam and the horizontal velocity derived from the vertical and a radial beam (r+e) to give the momentum flux in the vertical plane. defined by the two beams. -- t uw, w'(r+, - w'cose) = sin B where w'represents the vertical beam measurement. Other combinations of beams can, in theory be used (Worthington and Thomas, 1996). A third possible method. a 'hybrid' of equation (3.3) and equation (3.4), measures the perturbations of vertical wind directly and those of the horizontal wind usins a pair ol radial beams: -,, UW' = wr(r+@ - c0) 2 sin.g All these methods are used to calculate the momentum fluxes of gravity waves. The present work uses a direct approach where the time series of the

5 perturbation components of u and w are obtained using a suitable filtering of the original time series for isolating a particular frequency or a frequency band of interest. Then products u hi are formed and averaged over a suitable length of time Estimation of Ve1ocit:y Components For this study the time series of zonal wind (u) and vertical wind (w) are made use for identifying various equatorial wave modes and computing the associated momentum flux. Daily Doppler power spectra are collected from Indian MST radar located at Gadanki (13.5"N. 79.2"E) and zonal, meridional and vertical components are derived from the Doppler spectra. Observation of winds in the 3-20 km height region is made. 'The duration of observation is -10 minutes every day at hours. For this observation the following radar configuration was used: pulse repetition frequency = 1000 cs-'; pulse width = 16 ps coded with baud length of 1 ps; number of coherent integration = 128; number of FFT points = 128; number of beams = 6 (two zenith be:uns and four 10" off zenith beams in the North, East, South and West directions). From the Doppler spectra the velocity components are derived as follows Incoherent Integration (Spectral Averaging) Four Doppler spectra are incoherently integrated in order to improve the detectability of the true Doppler peak. The advantage of incoherent integration is that it improves the detectability of a Doppler spectrum which is defined as where, P, is the peak spectral density of the signal spectrum, and o,,,is standard deviation of spectral densities. When the fluctuation of the signal spectral densities is much smaller than that of the noise. then equation (3.6) becomes

6 which is more commonly used as definition of the detectability (e.g., Gage and Balsley, 1978; Balsley and Gage, 1980). Incoherent integrati~on is achieved by averaging the power spectrum a number of times. where m is the number of spectra integrated. In this study m=4 The noise spectral density has a XZ distribution with 2 degrees of freedom, because the noise spectral density is a summation of the squares of the real and imaginary components of the amplitude spectrum, which are assumed to have a Gaussian distribution. Therefore the mean value p and standard deviation o of the distribution is obtained as For a single spectrum oh, is equal to P,. When Doppler spectra are integrated incoherently by averaging m times, the mean values of the spectral densities of both the signal and noise are not changed. But o,j P, becomes lldm according to equation (3.7), because m incoherent integration of the noise provides a xz distribution with 2m degrees (of freedom. As a result D is increased by dm Power Spectrum Cleaning Constant frequency bands will form in the power spectrum by the interference generated in the system or due to extraneous signal. Due to this reason it is also possible for the fornlation of multiple bands in spectrum. This is removed by taking a range bin, which does not have echoes but the interference. This range bin is subtracted from all other range bins for the removal of mean noise.

7 Noise Level Estimation The observed Doppler spectra are invariably contaminated by radar system noise. It is important to know the spectral power density below which the spectrum is dominated by the instrument noise rather than by the signal received from the atmospheric target. This spectral power density is referred to as the noise threshold. Noise level is determined by the method developed by Hildebrand and Sekhon (1974). The noise level threshold is estimated to the maximum level L, such that the set of spectral points below the level S, nearly satisfies the criterion, variance (S)/mean(S) 2 1 over number of spectra averaged. The noise level is estimated as explained below. follows R, is a criteria to check the distribution of noise power and is defined as where S,, is the spectral density, and (S,,) and var(s,j are mean and standard deviation of the spectral density, respectively. The spectral densities are sorted in increasing order and eliminated one by one until R, = 1. (Petitdidier ef al., 1997) Moments Estimation From the daily Doppler spectra corresponding to the six beams with a vertical resolution of 150 m, the zeroth, first and second moments are calculated which correspond to the total power, the frequency shift and the spectral width respectively. In MST radars the radar signals are obtained mostly from turbulent scattering. In the case of scattering from turbulence induced fluctuations, the distribution of velocities in the turbulent scatter volume is Gaussian, and 'consequently the shape of Sfl is also Gaussian. A Gaussian and stationary process is

8 fully characterised by its autocorrelation function p(z), or equivalently by its Fourier transform, the frequency power spectrum Sfl. A Gaussian power spectrum has the form where f is the radian frequency. The power spectrum is fully defined by the three parameters P, x,, and u. They correspond to the total power, the frequency shift and the spectral width respectively. Therefore, if the spectrum is Gaussian, these three parameters contain all the i~lformation we can obtain from the radar echoes and they all should be known to know to characterise the process. They are a measure of three important physical properties of the medium: turbulence intensity, mean radial velocity and velocity dispersion. These three parameters correspond, also, to the first three moments of SV) Processing techniques for the estimation of the power, the frequency shift and the spectral width of MST radar returns are presented in Woodman (1985) Doppler Effect - Line of Sight Velocities In a given frame of reference if there is relative motion between the target and radar as the target encounters the electromagnetic waves radiated by radar, the frequency measured at the target will be different from that transmitted by radar. This is known as the Doppler effect. This frequency of the backscattered signals will again undergo Doppler shift as it reaches the radar. The Doppler frequency shift of echoes from moving target relative to the radar is given by

9 where v, is the line-of-sight (LOS) components of velocity vector v of the target relative to the radar.,fr<&,,, i:r the frequency of the electromagnetic wave and c is the velocity of electromagnetic wave.. The six line-of-sight (LOS) components of velocity is calculated from the Doppler frequency using the equation (3.15) Dimensional Winds From the daily venical profiles of LOS winds corresponding to the six beams, the zonal (u). meridllonal (v) and vertical (w) components of the daily winds were derived using a least-squares technique which is discussed as follows. The Doppler beam swinging method makes use of multiple antenna beams each of which is oriented to observe the radial velocity, at a different direction. The velocity vector is computed from the line-.of-sight velocities from these directions. Here it is assumed that the velocity field is uniform in space over the volume, which contains the range cells used to compute a velocity vector. In the atmospheric radar application, it is common to determine a velocity vector from line-of-sight velocities of range cells with the same height assuming the uniformity only in the horizontal plane, so that a height profile of the velocity vector can be obtained. This is because the horizontal velocity is usually much larger than the vertical velocity in the stratified earth's atmosphere, thus making the horizontal uniformity of the velocity field much better than in the vertical direction. The line-of-sight component of the wind velocity vector v = (v,, v,, v:) at a given height is expressed as where i is a unit vector along the antenna beam direction, and 0,, q,, 0, are the angle between i and the x, y and z axes, respectively. If we measure L:, at In beam directions, then the estimate of v can be determined in a least-square manner, with which the residual given by the following is minimised.

10 The necessary condition for c,.' to give the minimum is that partial derivatives of with respect to all the three components of v are zero. This set of equations can be solved in terms of v as where the summations are taken for I = 1 to m (Sato, 1989). On solving this equation we can derive L'~, 1% and v. which correspond to u (zonal), v (meridional) and w (vertical) components of velocity Once the vertical profiles (height resolution of 150 m) of u, v, and w are obtained, these are subjected to a 5 point running mean in height with weights 0.125, 0.125, 0.5, andl This smoothing procedure will reduce the small scale wind fluctuations with height which may be due to random fluctuations or real atmospheric phenomena. Homwever, the interested wave components for this study are the equatorial waves whose vertical wavelengths are in the range of 4-12 km. Time series of u and w are obtained from height-smoothed wind profiles. There exists two or three gaps of one or two days in the time series, which are, filled by linear interpolation from adjacent wind values. The length of the time series is chosen as 48 days. This data length is sufficient for the study of slow Kelvin (10-20 day period) and RG waves (3-5 day period). The time series of u, v, and w can be spectrum analysed to bring out the various periodic components present in the data and Kelvin and RG waves may be identified by their periods. Before doing that it is worthwhile to examine whether these Kelvin and RG waves can be expected to be present with sufficient amplitudes for detection in the wind data obtained at a location 13.5" north of the equator.

11 Theoretically, Kelvin waves will have maximum zonal wind amplitude over the equator and their amplitude:; will fall off exponentially on both sides of the equator, whereas RG waves will have maximum amplitudes near 8" North and 8" South of the equator and their amplitudes will fall off exponentially both equatorwards and polewards (Holron. 1980). The expected zonal wind amplitudes of the slow Kelvin and RG waves for a latitude of 13.5" N are -2.5 m s-' and m s-' respectively. The estimated random erroi: in the radar-measured winds is -&0.1 m s-l (based on the Doppler spectral resolution). Thus both Kelvin and RG waves with these amplitudes can be detected trom the radar wind data obtained at Gadanki Present Method to Estimate the Equatorial Wave Momentum Fluxes The vertical flux density of horizontal momentum is a two-dimensional vector quantity that describes the direction of the horizontal momentum that is being advected vertically and the rate at which it is being advected. In general, the - - components of this vector are denoted pu'w' and pv'w' where p is the density (the overbar denotes a perturbation about a mean). The total wind velocity vector can be represented by L = (; +U' where C' = (u', v', w') is the instantaneous perturbation about that mean. The time average that is inherent in this linearisation process has the effect of excluding contributions due to periodic variations with timescales that are significantly longer than averaging interval. These longer-term variations contribute to the mean in s~uccessive averaging intervals but are not represented in the perturbation quantities. Thus the linearisation process partitions the velocity variations by frequency suc:h that higher-frequency components (from the Nyquist frequency down to approxiniately the inverse of the sampling interval) are present in the time series of the perturbation velocity vector U', while lower frequency components remain in (;. In turn, the vertical flux density of horizontal momentum described by pu'm;' and p.9'~' is due to velocity variations whose frequencies are similarly defined by the averaging interval and the linearisation process. The momentum flux densities of two waves of the same frequency can be summed if the waves are independent of each other.

12 The perturbations u 'and w'are in phase (time) at a particular height (Holton, 1980) for Kelvin and RG waves and this temporal coherence will be reflected in the cross spectrum of u and u. In order to see the coherence between u'and w' the cross-spectrum of u and w is estimated using the following fonnula Here S,, V) is the cross-spectrum; Ufl and WV) are the Fourier transforms of u(t) and w(t) respectively. W*fl is the complex conjugate of WV); f and t are frequency and time respectively. The angle bracket indicates ensemble averaging. The time series of u and w are available every 150 m and the ensemble average is done in the height domain over six consecutive heights, thus yielding a cross-spectrum S at every 900 m. The magnitude of 'S' is the coherence, which can vary from 0 to 1; it gives a measure of the correlation between u and w in a particular spectral bin. Since u and w vary in time in a non-random manner for the equatorial waves, a high coherence value indicates the presence of wave motion corresponding to that frequency. Similarly the cross spectrum of u and v is obtained. Since the Kelvin waves have no rneridional component there need not be any correlation between u and v. But for RG waves there will be good correlation between u and v. It will be interesting tc) see whether the coherent oscillations at the prominent periods belong to any propaagating (in the vertical) waves. This can be done by examining the vertical coherence of the oscillations in the zonal wind component (u) at the prominent periods. For this purpose a reference height is chosen in the upper troposphere and selection of the height is based on the high coherence (between u and w) values. Then the cross-spectrum of the zonal winds (u) and the vertical coherence at all heights relative to these reference heights were found applying a similar method as used in con~puting the cross-spectrum of u and w. If the vertical coherence value 1s also high for the prominent periods obtained in the cross spectrum, one can reasonably conclude that high coherence between u and w is probably due to the propagation of atmospheric waves in the vertical direction.

13 Though periodicities alone can not identify the different equatorial wave modes with certainty. their vertical propagation characteristics also can yield useful information for identifying different modes. For this purpose, the vertical wavelengths are estimated corresponding to the prominent periodic components from the vertical profile of phase (time of maximum) of these components obtained from the autospectrum of the zonal wind u. The autospectrum was obtained as follows. First the time series of u were averaged over six consecutive heights to yield time series at every 900 m which were then subjected to spectral analysis. This averaging is for making the height resolution of the auto- and cross- spectrum identical. The amplitude spectrum of u and v are obtained at every 900 m. From these spectra we can get an idea of prominent periods in the time series and the altitude at which they are prominent. The corresponding amplitude profie of u is also calculated. Since the middle troposphere is believed to be the source region for these waves (Andrrws et a/., 1987), the vertical wavelengths may be better estimated above 10 km altitude (i.e. the propagation region). The observed amplitudes and vertical wiivelengths are compared with the theoretically expected values. It may be noted that the expected vertical wavelengths of slow Kelvin waves and RG waves are 6-12 km and 4-8 krn respectively, and the corresponding expected zonal wind amplitudes for 13.5"N latitude are -2.5 and 1.25 m s-i, respectively (Holron, 1980). If both the observed and theoretical values of amplitudes and vertical wavelengths are matching with each other it is reasonable to conclude that the prominent periods observed in the cross spectrum of u and w and vertical coherence spectrum are due to equatorial waves. Thus equatorial Kelvin and RG wave modes were identified during different seasons. Now the vertical nlux of zonal momentum due to the identified prominent equatorial waves are calculated. First the amplitudes and phases of different periodic components at 900 m an: obtained as discussed earlier. The filtering of the time series is done by reconstructing the series separately for each prominent period, using the corresponding Fourier amplitudes and phases. Then products u'w'are - formed and averaged over the selected length of time. Thus u'w' is evaluated separately for each period.

14 3.6. Accuracy of Momentum Flux Estimates - A short discussion about the errors in the computed values of u'w' will be in order. For a rigorous evaluation of the errors evolved, the signal-to-noise ratio (SNR) of the radar echoes which yield the LOS Doppler shift has to be taken in to account. However, we will derive the errors based on the assumptions that the observed u and w correspond to a high SNR and that beam width effects, errors due to aspect sensitivity etc. can be neglected so that half of the Doppler resolution can be taken as the error in u and w. Here the spectral resolution is Hz, or 0.18 m s-i, so that error is _ m s (assumed to be random). Four consecutive Doppler spectra obtained from each radar beam in each range bin are averaged to improve the detectability of the signal (by reducing the variance of the noise) and this process is referred to as incoherent averaging. Four incoherent averaging and averaging over six height levels reduces the error to a = 0.02 m s-i. The random error a, in the wind components wil:l cause a random error of - a in the estimated amplitudes, when using spectral (harmonic) analysis. Here n = 48, the number of samples used for determining the spectral amplitudes. Thus the spectral analysis of the time series results in ail error of A m s-' in the estimated amplitude of the spectral component. Assuming the error in the reconstructed time series of u - and w to be o* = crw '0.004 m s-i, the error in the estimated u'w' becomes (neglecting the second term) Putting a typ~c;il value of u=2 m s-' (approximately the amplitude of the zonal wind oscillat~on), we get c,, = 1.2x104 m's-'. Thus it appears that the method used to estimate u'w' in this work will give meaningful values for the momentum flux of the equatorial waves. In addition to these errors, geophysical noise present in the radar wind data produces statistical uncertainty in the estimated momentum flux values depending on the magnitude of atmospheric momentum flux ( w ) relative to the geometric mean energy ( ( t W 2 ( )) (Kudeki and

15 Franke, 1998). Accordingly Kudeki and Franke (1998) suggest long integration times for statistically significant estimates of the momentum flux. Though the concept developed by them is mostly applicable to gravity wave momentum flux estimates (based on a particular spectral distribution of energy) the idea of using a long integration time is equally applicable for other wave motions such as equatorial waves also. An order of calculation of ratio of momentum to the geometric mean energy in our case gives a value of -10% and we have used an integration time of 48 days which may be sufficient for a statistically reliable estimate of the momentum flux. However, estimated r.m.s. error in momentum flux u'w' given by the above discussed method could be a lower limit in the context of possible errors introduced by geophysical noise Comparison with Other Methods In the earlier studies reported, the momentum fluxes of equatorial waves were estimated using radiosonde data (e.g., Wallace and Kousky 1968b, Sato and Dunkenon, 1997). These are in situ measurements which themselves will disturb the medium. The radiosondes are making point measurements which may not be representative of the atmospheric conditions averaged over space whereas radar provides volume averages. In the Wallace and Kousky (1968b) method vertical wind perturbations indirectly obtained from temperature fluctuations are taken. In Sato and Dunkerron (1997) the horizontal momentum flux were calculated from quadrature spectra of zonal wind and temperature. In this method also estimates of momentum flux is based on similar principles used by Wallace and Kousky. In the present method the zonal and vertical components of wind velocity measured by MST radar are used to estimate the momentum fluxes of equatorial waves. Vincent and Reid (1983) method for calculation of gravity wave momentum fluxes were based on so many assumptions. One of the assumptions made in the calculations is that the Doppler velocities were measured along the beam axes. If the irregularities responsible for radiowave backscatter were not isotropic in their characteristics. then the effective orientation would not be along the beam axes. To use Vincent and Reid (1983) method, it is necessary to make the assumption that the

16 statistical properties of the atmosphere at one height are spatially invariant in a horizontal plane. It is not reasonable to assume that the instantaneous velocities in the two sampling volumes are the same. In the present developed method no such assumptions are made. The Vincent and Reid (1983) method of estimating momentum flux involves the difference between two quantities, which are similar in magnitude. The quantities chat are being differenced are variances and these are susceptible to the effects of outliers. The effects of much noisier data in one of the symmetric beam will increase the variance in that beam. Any increase of variance in one beam implies stronger wave propagation near the horizontal direction corresponding to that beam, Hence differences in data quality between beams lead to false estimates of wave momentum flux. It is therefore important to remove the outliers from the data. In this method the magnitude of the momentum fluxes depends on an accurate estimate of the beam-pointing angle 8. It was suggested by Vincent and Reid (1983) that their momentum flux measurements underestimated the true momentum flux because the effective pointing angle of the radar beams was closer to the zenith than that defined by the antenna phasing. It may be mentioned here that this method makes use of u and w obtained from the six LOS wind values, where as Vincent and Reit1 method makes use of only two beams Summary A method to estimate the vertical flux of zonal momentum of the equatorial waves from MST radar measured winds in the tropical troposphere and lower stratosphere is discussed. This method involves identifying prominent equatorial wave (Kelvin and RG) modes in the data from cross spectrum of zonal and vertical winds and vertical coher~x~ce spectrum of zonal wind. To ensure that the identified periods are due to equatorial waves, their vertical propagation characteristics (amplitudes and vertical wavelengths) are also studied and compared with theoretical values. Then the filtered time series of zonal and vertical winds corresponding to the identified prominent periods are obtained. Using these time series momentum flux is computed involving a direct method. An estimate of the errors in the computed flux values in the present estimate gives a value of -1.2 x 10.' mzs r$ ---