16.1 Introduction

Internal gravity waves that we discussed in the previous chapter are short-period small-scale waves which are mainly due to buoyancy and the propagation of which is influenced by gravity. There are, however, some longer-period larger-scale waves in the atmosphere the propagation of which is affected by not only gravity but also the earth's rotation. These are called inertia-gravity waves. However, in the case of such waves, it can be shown that not all can propagate vertically in the atmosphere. The criterion for such vertical propagation is that the frequency of the waves must be greater than the Coriolis frequency. In midlatitudes, where the Coriolis frequency is large, many low-frequency long-period planetary-scale waves fail to meet this criterion and the waves are, so to say, trapped by the atmosphere. But the situation is different near the equator where there is little Coriolis control and the decreasing Coriolis frequency allows these long-period planetary-scale waves to be untrapped and move vertically. According to theory, these long-period vertically propagating waves generally must have very short vertical wavelengths.

There are, however, two types of vertically propagating waves whose wavelengths are large enough to be easily detected. These are the eastward-propagating Kelvin wave, and the westward-traveling mixed Rossby-gravity wave.

The Kelvin wave has a distribution of pressure and velocity which is symmetric about the equator and has little meridional velocity, whereas the mixed Rossby-gravity wave has a distribution of pressure and velocity which is antisymmetric about the equator but has a distribution of meridional velocity which is symmetric about the equator.

The horizontal distributions of pressure and velocity characteristic of these waves are shown in Fig. 16.1.

The other interesting low-frequency equatorial waves and oscillations that we examine in this chapter are the Quasi-Biennial Oscillation (QBO), the Madden Julian Oscillation (MJO) and the El Nino Southern Oscillation (ENSO). It is felt that a familiarity with tropical circulation systems and their properties will facilitate an understanding of these waves.

The dynamics of all these waves and oscillations can be deduced theoretically by using the log-pressure co-ordinate system in the governing equations and applying the linear perturbation technique introduced in Sect. 15.5. An introduction to the waves is furnished here, following a treatment given by Holton (1979).

16.2 The Governing Equations in Log-Pressure Co-ordinate System

In the log-pressure co-ordinate system, the vertical height z* is defined by where H is the scale height given by H = RT0/g with T0 a global average temperature, and p0 is a standard reference pressure usually taken to be 100 kPa. For an isothermal atmosphere at temperature T0, z* is exactly equal to the geometric height. For a variable temperature, they are only approximately equal. In this system, the vertical velocity w* is given by z*^ Hln(p0/p)

The governing equations in the log-pressure system are given by:

16.2.1 The Horizontal Momentum Equations dV/dt + f kxV = - VO

where the operator d/dt is now defined as d/dt = d/dt + V. V + w*d/dz:

16.2.2 The Hydrostatic Equation dO/dz* = RT/H (16.2.4)

which is obtained with the aid of the ideal gas law (2.4.12).

16.2.3 The Continuity Equation

The continuity equation in the log-pressure system is obtained by transforming from the isobaric co-ordinate system as follows:

In the isobaric co-ordinate system, du/dx + dv/dy + dm/dp = 0, where m is the vertical p-velocity(= dp/dt). Since w* = dz*/dt = —Hm/p, we can write dm/dp = —d(pw*/H)/dp = dw*/dz — w*/H

Thus, in the log-pressure system, the continuity equation is du/dx + dv/dy + dw*/dz* — w*/H = 0 (16.2.5)

The thermodynamic energy equation (11.9.1) in isobaric co-ordinates dQ/dt = cp dT/dt — a dp/dt when transformed into log-pressure co-ordinates, with the aid of (16.2.2), (16.2.3) and (16.2.5), takes the form

In the stratosphere, N2, the buoyancy frequency squared, is approximately constant with a value around 4 x 10—4 s—2.

We now useEqs. (16.2.3.), (16.2.4), (16.2.5) and (16.2.6) referring them to an equatorial P plane with the Coriolis parameter approximated by f = 2 Q y/a = Py where y is the distance from the equator and a is the mean radius of the earth. This means that P = df/dy.

We assume a basic state of the atmosphere which is at rest (Note that adding a constant zonal velocity simply Doppler-shifts the frequency) and has no diabatic heating.

The perturbations are assumed to be zonally propagating waves and may be written as u = u/(y, z* ) exp (i (kx - |t))

w* = w*'(y,z*) exp(i (kx-|t)) O = O' (y, z*) exp(i (kx - |t))

where k and | are respectively the wave number and frequency of the wave .

Substituting (16 . 3 .1) in (16 .2 . 3, 16 .2 . 5 and 16 . 2 . 6) and linearizing, we get the perturbation equations

- i|v' + Pyu' = -dO'/dy (16.3.3) iku' + dv'/dy + (d/dz* - 1/H)w*' = 0 (16.3.4)

Since, in a Kelvin wave, there is no meridional velocity, we put v' = 0 and eliminate w*' between (16.3.4) and (16.3.5) to get

We use (16.3.6) to eliminate O' from (16.3.7) and (16.3.8) and get two independent equations which the field of u' must satisfy. These are:

Equation (16.3.9) determines the meridional distribution of u' and (16.3.10) determines the vertical distribution.

It is easily verified that (16.3.9) has the solution u' = uc(z*)exp{- (pk/2|)y2} (16.3.11)

If we assume that k > 0, then | > 0 corresponds to an eastward propagating wave the amplitude of which is maximum at the equator and falls off exponentially with y on either side. The field of u' in that case has a Gaussian distribution about the equator with an e-folding width given by

If we had taken | < 0, it would have resulted in a westward moving wave the amplitude of which would increase exponentially away from the equator. However, such a result would violate reasonable boundary conditions at the poles and must, therefore, be rejected. Thus, there exists only an eastward propagating atmospheric Kelvin wave.

According to Holton and Lindzen (1968), Kelvin waves may be defined as shallow water gravity waves which propagate parallel to a coastline and have no velocity component normal to the coastal boundary. The latter condition implies that the pressure gradient normal to the coastline be in geostrophic balance with the velocity field, which in turn requires that the amplitude of the wave decay exponentially away from the coast. In view of the similarity of the present solution (16.3.11) to Kelvin waves, it seems reasonable to call the wave an atmospheric "Kelvin" wave, noting that the equator plays the same role as a coastal boundary.

The solution of (16.3.10) which gives the vertical structure of the wave may be written in the form u'(z*)=uo(z*) exp (z*/2H){Cx exp(i^z*) + C2 exp(-i^z*)} (16.3.13) where = (N2k2/|2) - (1/4H2), and Ci, C2 are constants which are to be determined from appropriate boundary conditions.

For > 0, the solution (16.3.13) is in the form of a vertically propagating wave and identical to an eastward propagating internal gravity wave that we discussed earlier in Sect. 15.7.1. Its eastward phase velocity has a downward component; hence the constant C1 in (16.3.13) must be zero.

However, its group velocity, i.e., the direction of energy propagation, has an upward component. The Kelvin wave thus has a structure in the x, z plane which is shown in Fig. 16.2, which is identical to that shown earlier in Fig. 15.7.

We can make a similar analysis for the mixed Rossby-gravity wave. But, for this case, we must use the full perturbation equations (16.3.2-16.3.5). The solution corresponding to the pattern shown in Fig. 16.1(b) is

(v' )=¥(z*)(1) exp[{-(1 + k|/p)p2y2}/2|2] (16.4.1)

Fig. 16.2 Longitude-height section along the equator showing pressure, temperature and wind perturbations for a thermally damped Kelvin wave. Heavy wavy lines indicate material lines, short blunt arrows show phase propagation. Areas of high pressure are shaded. Length of small thin arrows is proportional to the wave amplitude which decreases with height due to damping. The large shaded arrow indicates the net mean flow acceleration due to the wave stress divergence

Fig. 16.2 Longitude-height section along the equator showing pressure, temperature and wind perturbations for a thermally damped Kelvin wave. Heavy wavy lines indicate material lines, short blunt arrows show phase propagation. Areas of high pressure are shaded. Length of small thin arrows is proportional to the wave amplitude which decreases with height due to damping. The large shaded arrow indicates the net mean flow acceleration due to the wave stress divergence where the vertical structure ¥(z*) of the three variables is given by

¥(z*) = exp(z*/2H){Ci exp(iV*)+C2 exp(-iV*)} (16.4.2)

where ^g = (N2 k2/M)(1 + PAm)2 - (1/4H2), and C1, C2 are constants which are to be determined by the boundary conditions.

Equation (16.4.1) shows that the mixed Rossby-gravity wave mode vf has a Gaussian distribution about the equator with an e-folding width given by

Equation (16.4.3) is valid for westward propagating waves (m < 0) provided that

(1 + Mk/P) > 0, which condition can also be written in the form

In the case of frequencies which do not meet the condition (16.4.4), the wave amplitude will not decay away from the equator and hence will not satisfy boundary conditions at the pole.

For upward energy propagation, the mixed Rossby-gravity waves must have downward phase propagation just as in the case of Kelvin waves. Thus, the constant C2 in (16.4.2) must be zero. The resulting wave structure in the x, z plane at a latitude north of the equator is shown in Fig. 16.3.

Fig. 16.3 Zonal-vertical (x, z) section along a latitude circle north of the equator showing the distribution of pressure, temperature and wind perturbations for a thermally damped mixed Rossby-gravity wave. Areas of high pressure are shaded. Small arrows indicate zonal and vertical wind perturbations with length proportional to wave amplitude. Meridional wind perturbations are shown by arrows pointed into the page (northward) and out of the page (southward). The large shaded arrow shows the direction of net mean flow acceleration due to the wave stress divergence

Fig. 16.3 Zonal-vertical (x, z) section along a latitude circle north of the equator showing the distribution of pressure, temperature and wind perturbations for a thermally damped mixed Rossby-gravity wave. Areas of high pressure are shaded. Small arrows indicate zonal and vertical wind perturbations with length proportional to wave amplitude. Meridional wind perturbations are shown by arrows pointed into the page (northward) and out of the page (southward). The large shaded arrow shows the direction of net mean flow acceleration due to the wave stress divergence

It is interesting to note from Figs. 16.1(b) and 16.3 that the mixed Rossby-gravity wave removes heat from the equatorial region to higher latitudes, since the poleward moving air is correlated with positive temperature perturbations so that the eddy heat flux (vfTf) is positive.

Both Kelvin and mixed Rossby-gravity wave modes have been identified in observations from the equatorial stratosphere. The observed Kelvin wave appears to have a period in the range 12-20 days, a zonal wave number 1, (i.e., one wave appears to span the whole latitude circle) and a phase speed of about 30m s—1 relative to the ground. If we assume a mean zonal wind of — 10ms—1 so that the Doppler-shifted phase velocity c ~ 40ms—1, then we find from (16.3.12) that YL is about 2000 km. This agrees well with the observation that the significant amplitude of the Kelvin wave is largely confined within about 20° of latitude of the equator. Further, knowledge of the phase speed allows us to compute the vertical wavelength of the Kelvin wave. Assuming a value N2 = 4 x 10—4 s—2 and using (16.3.13), we get vertical wavelength ^ 2n/X ~ 2nc/N ~ 12 km, which is in good agreement with the value of the vertical wavelength deduced from observations. An example of zonal wind oscillations caused by passage of Kelvin waves at Kwajalein, a station near the equator in western north Pacific, is shown by a time-height section in Fig. 16.4.

The existence of mixed Rossby-gravity mode has also been confirmed in the observational data over the equatorial Pacific. This mode is most easiliy identified

MAY JUJY JUL AUG

1963

Fig. 16.4 Time-height section of zonal wind at Kwajalein (near 9N).Isotachs at intervals of 5 m s-1. Westerlies are shaded. (After Wallace and Kousky, 1968) (Reproduced with permission of the American Meteorological Society)

MAY JUJY JUL AUG

1963

Fig. 16.4 Time-height section of zonal wind at Kwajalein (near 9N).Isotachs at intervals of 5 m s-1. Westerlies are shaded. (After Wallace and Kousky, 1968) (Reproduced with permission of the American Meteorological Society)

by the meridional wind component, since, according to (16.4.1), vf is a maximum at the equator (y = 0) for the mixed Rossby-gravity mode. The observed waves of this mode have periods in the range 4-5 days and propagate westward at about 20m s-1. The horizontal wavelength appears to be about 10,000km. The observed vertical wavelength is about 6 km which agrees closely with the theoretically derived wavelength from (16.4.2). These waves also appear to have significant wavelength within about 20o of the equator, which is consistent with the e-folding width YL derived from (16.4.3).

Theory and observations appear to suggest that both the Kelvin wave and the Rossby-gravity wave are excited by oscillations in the large-scale convective heating pattern in the equatorial troposphere. Though these waves do not contain much energy compared to other tropical disturbances such as storms and cyclones, they are the predominant disturbances of the equatorial stratosphere, and through their vertical energy and momentum transport play an important role in the maintenance of the general circulation of the stratosphere.

16.6 The Quasi-Biennial Oscillation (QBO)

Among the various types of atmospheric oscillations with which we are by now familiar from observations, the quasi-biennial oscillation of the zonal wind in the equatorial stratosphere is, perhaps, closest to one exhibiting true periodicity. It has the following observed features: Zonally-symmetric easterly and westerly wind regimes appear alternately with a period between 24 and 30 months. Successive regimes first appear above 30 km, but their phases propagate downward at a rate of about 1 km per month. The downward propagation occurs without change of amplitude between 30 and 23 km but there is rapid loss of amplitude below 23 km. The oscillation is symmetric about the equator with maximum amplitude of about 20m s_1 and a half-width of about 12° of latitude. It is well depicted by a time-height section of mean zonal wind components (m s_1) for Canton Island during the period February 1954 to October 1960, as shown in Fig. 16.5. One may note from Fig. 16.5 that the vertical shear of the wind is quite strong in the region where one regime is replacing the other. Since the oscillation is zonally symmetric and symmetric about the equator and has very small meridional as well as vertical motions associated with it, the zonal wind must be in geostrophic balance nearly all the way to the equator. Thus, there needs to be a strong meridional temperature gradient in the vertical shear zone to satisfy the thermal wind balance.

Several observed features of the quasi-biennial oscillation, such as its approximate biennial period, downward phase propagation without loss of amplitude, the occurrence of zonally symmetric westerlies at the equator, require theoretical explanations and factors responsible for them need to be identified. The occurrence of westerlies at the equator signifies that air is moving eastward faster than the earth's surface and, therefore, cannot be accounted for by advection of angular momentum from higher latitudes where the earth's angular momentum is less than that at the equator. So, there must be some eddy momentum source to create the westerly accelerations in the downward-moving westerly shear zone.

Both observational and theoretical studies have confirmed that vertically propagating equatorial waves - the Kelvin wave and the mixed Rossby-gravity waves -provide the zonal momentum sources needed to drive the quasi-biennial oscillation. Fig. 16.2 shows that eastward-propagating Kelvin waves with upward energy propagation transfer westerly momentum upward, i.e., u'w' > 0. This means that u' and w' are positively correlated in the case of these waves. Fig. 16.3 shows that u'w' > 0, also in the case of mixed Rossby-gravity waves. However, the mixed Rossby-gravity

FEa JUN OCT FEB JUN OCT FEB JUN OCT FEB JUN OCT FEB JUN OCT FES JUN OCT FEB JUN OCT !954 (955 1956 1957 f95S 1959 I960

Fig. 16.5 Time-height section for Canton island (2°46'S, 171°43'W), February 1954-October 1960. Isopleths are monthly mean zonal wind components inms"1. Negative values denote easterly winds (After Reed and Rogers, 1962 Reproduced with permission of the American Meteorological Society)

FEa JUN OCT FEB JUN OCT FEB JUN OCT FEB JUN OCT FEB JUN OCT FES JUN OCT FEB JUN OCT !954 (955 1956 1957 f95S 1959 I960

Fig. 16.5 Time-height section for Canton island (2°46'S, 171°43'W), February 1954-October 1960. Isopleths are monthly mean zonal wind components inms"1. Negative values denote easterly winds (After Reed and Rogers, 1962 Reproduced with permission of the American Meteorological Society)

mode has a strong positive horizontal heat flux, v/T/ > 0, which sets up a mean meridional circulation which through the Coriolis acceleration fv' < 0 produces a net easterly acceleration.

In this context, the two heavy wavy lines shown in Figs. 16.2 and 16.3 are highly significant. If we take the lower line as representing the vertical displacement of material particles at the tropopause level at different longitudes, the upper line shows that the amplitude of the displacement decreases with height in the stratosphere.

Theoretical calculations indicate that the equatorial stratospheric waves are thermally damped by infrared radiation to space so that the amplitude of the temperature wave decreases with height. Further, the extent of damping depends strongly on the doppler-shifted frequency of the waves. As the doppler-shifted frequency decreases, the vertical component of the group velocity also decreases, so a much longer time is available for the energy to be damped for a rise through a given vertical distance. Thus, the westerly Kelvin waves tend to be damped rapidly in westerly shear zones below their critical levels (a critical level is the altitude at which the relative horizontal phase speed of an internal gravity wave equals the mean wind speed). As a wave approaches this critical altitude, from below or above, the vertical component of the group velocity becomes zero and its energy is absorbed and transferred to the mean flow. This causes the westerly shear zone to descend with time. Similarly, the easterly mixed Rossby-gravity waves are damped in easterly shear zones, thereby causing an easterly acceleration and lowering of the easterly shear zone.

Thus, we may conclude that the quasi-biennial oscillation is, indeed, excited by the vertically propagating equatorial waves through the mechanism of radiative damping which causes the waves to decay in amplitude with height and thus transfer their energy to the mean zonal flow.

16.7 The Madden-Julian Oscillation (MJO)

This oscillation, named after its co-discoverers, Madden and Julian(1971, 1972), is a broad band intraseasonal tropical oscillation which was first detected in the co-spectrum of the 850- and 150-mb zonal wind (u) components in which a very pronounced negative coherence extreme was noted in the frequency range 0.02450.0100 day-1 (period 41-53 days). Their analysis based on nearly ten years of daily rawinsonde data for Canton Island (3S, 172W) revealed the following structure of this oscillation in the fields of wind, pressure and temperature:

(a) In the wind field, peaks in the variance spectra of the zonal wind were found to be strong in the lower troposphere, weak or non-existent in the 700-400 mb layer, and strong again in the upper troposphere. No evidence of this feature could be found above 80 mb, or in any of the spectra of the meridional component.

(b) In the pressure field, the spectrum of station pressure showed a peak in this frequency range and the oscillation was in phase with the lower-tropospheric zonal wind oscillation but out of phase with that in the upper troposphere.

(c) The tropospheric temperatures exhibited similar peak and were highly coherent with the station pressure oscillation, positive station pressure anomalies being associated with negative temperature anomalies throughout the troposphere. Thus, the lower-middle troposphere appears to act as a nodal surface with u and surface pressure oscillating in phase, but 180° out of phase above or below.

Their study (Madden and Julian, 1972) also appears to indicate that the detected oscillation is of global scale but restricted to the tropics and is the result of an eastward movement of large-scale circulation cells oriented in the zonal plane. Fig. 16.6 shows (upper panel) the co-spectrum between the 850- and 150-mbu components, along with the co-spectrum between the station pressure and the 850-mb u component, as originally presented by Madden and Julian (1971). In both the co-spectra, one can see peaks in the frequency range of the MJO.

Fig. 16.6 (Top) The co-spectrum of the 850- and 150-mb zonal wind u(dashed, and left ordinate values), and co-spectrum of the station(sfc) pressure and the 850-mb zonal wind(solid, and right ordinate values) for Canton Island, June 1957 through March 1967. The ordinate is co-spectral density normalized to unit bandwidth in unit of (m2 s-1 day-1). (Bottom) The coherence-squared statistic for the 850- and 150-mb zonal wind and the station pressure and 850-mb u -series (Reproduced from Madden and Julian, 1971, with permission of American Meteorological Society)

Fig. 16.6 (Top) The co-spectrum of the 850- and 150-mb zonal wind u(dashed, and left ordinate values), and co-spectrum of the station(sfc) pressure and the 850-mb zonal wind(solid, and right ordinate values) for Canton Island, June 1957 through March 1967. The ordinate is co-spectral density normalized to unit bandwidth in unit of (m2 s-1 day-1). (Bottom) The coherence-squared statistic for the 850- and 150-mb zonal wind and the station pressure and 850-mb u -series (Reproduced from Madden and Julian, 1971, with permission of American Meteorological Society)

The Meteorological glossary of the American Meteorological Society (2000) describes a Madden-Julian oscillation as follows: It is "a quasi-periodic oscillation of the near-equatorial troposphere, most noticeable in the zonal wind component of the boundary layer and in the upper troposphere, particularly over the Indian ocean and the western equatorial Pacific. It appears to represent an eastward propagating disturbance with the structure of a Kelvin wave with a vertical half-wavelength of the depth of the troposphere, but with a phase speed of only about 8 ms-1, much less than that of an adiabatic Kelvin wave in the stratosphere. The disturbance is accompanied by strong fluctuations of deep convection, easily detectable using satellite observations, and is a major contributor to intraseasonal weather variability in equatorial regions from eastern Africa eastward to the central Pacific." As is well-known, this region of the equator is dominated by monsoons.

Both theoretical and observational studies during the last three decades have concentrated on an understanding of the mechanism of excitation of the MJO by deep convection and how its eastward propagation as a Kelvin wave is affected by convection. Madden (1986), using a technique which he termed seasonally varying cross-spectral analysis concluded that the energy source for the 40-50 day oscillation is the large-scale convection associated with the seasonally-migrating intertropical convergence zone (ITCZ). Madden found that the variance of the zonal wind in a relatively broad band centered on a 47-day period was maximum during December, January and February (DJF) and at stations in the Indian and western Pacific oceans during all seasons. The lower- and upper-tropospheric zonal winds tend to be most coherent and out-of-phase at near-equatorial stations in the summer hemisphere. It is likely that this results from the dependence of the oscillation on the deep convection associated with the seasonal migration of the ITCZ.

An interesting aspect of this oscillation, as found by his study, is the involvement of the meridional v-component of the wind in transferring mass from the convec-tive region of the summer hemisphere ITCZ to the winter hemisphere, as shown schematically in the upper part of Fig. 16.7.

Hendon and Salby (1994) constructed a composite life cycle of the MJO from the cross covariance between outgoing longwave radiation (OLR), wind and temperature, using episodes when a discrete signal in OLR is present. It was found that the composite convective anomaly possesses a predominantly zonal wavenum-ber 2 structure that is confined to the eastern hemisphere, propagates eastward at about 5ms-1 and evolves through a systematic cycle of amplification and decay. Unlike the convective anomaly, the circulation anomaly is not confined to the eastern hemisphere and travels faster. According to these workers, the circulation anomaly exhibits characteristics of both a forced response, coupled to the convec-tive anomaly as it propagates across the eastern hemisphere, and a radiating response which propagates away from the convective anomaly into the western hemisphere at about 8-10 ms-1. The forced response appears as a coupled Rossby-Kelvin wave, while the radiating response displays predominantly Kelvin wave features. During amplification, the convective anomaly is positively correlated to the temperature perturbation, which means production of eddy available potential energy

Fig. 16.7 Schematic of the structure of a large-amplitude 40-50 day oscillation in the equatorial plane The shading at the bottom of the lower panel represents the negative pressure anomalies.The line at the top of lower panel represents the tropopause. The upper panel is a plan view of the disturbance in the upper troposphere. The shaded area there corresponds to a positive anomaly in convection centered in the summer hemisphere. The divergence of the anticyclonic circulation transfers mass from summer to the winter hemisphere (Reproduced from Madden, 1986, with permission of American Meteorological Society)

Fig. 16.7 Schematic of the structure of a large-amplitude 40-50 day oscillation in the equatorial plane The shading at the bottom of the lower panel represents the negative pressure anomalies.The line at the top of lower panel represents the tropopause. The upper panel is a plan view of the disturbance in the upper troposphere. The shaded area there corresponds to a positive anomaly in convection centered in the summer hemisphere. The divergence of the anticyclonic circulation transfers mass from summer to the winter hemisphere (Reproduced from Madden, 1986, with permission of American Meteorological Society)

(EAPE). A similar correlation between upper tropospheric divergence and temperature implies conversion of EAPE to eddy kinetic energy during this time. When it is decaying, temperature has shifted nearly into quadrature with convection, so their correlation and production of EAPE are then small. The same correspondence to the amplification and decay of the disturbance is observed in the phase relationship between surface convergence and convective anomaly. The correspondence of surface convergence to the amplification and decay of the convective anomaly suggests that frictional wave-CISK plays a key role in generating the MJO.

A numerical study by Chang and Lim (1988) shows that two types of CISK modes may arise from an interaction of two vertical modes of convection, shallow and deep. With maximum convective heating in the lower troposphere, the instability is due to the lower internal mode which gives a stationary east-west symmetrical structure. However, when heating is maximum in the midtroposphere, eastward propagating CISK modes resembling the observed and numerically-simulated oscillations occur. These modes arise from the interaction of the two internal modes which are locked in phase vertically. A time-lagged CISK analysis suggests that the shallower mode, with its stronger influence on the low-level moisture convergence, slows down the deeper mode resulting in a combined mode which has a deep vertical structure but a relatively slow propagating speed.

16.8 El Niño-Southern Oscillation (ENSO)

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