1. Introduction

Eastward propagating Kelvin waves contribute prominently to the intraseasonal (i.e., periods shorter than a season) variability of the equatorial Pacific thermocline (e.g., Kessler et al. 1995, hereafter KMW). These waves are also detectable in equatorial sea level (Enfield 1987) and are traceable away from the equator as they propagate poleward along the western coast of the Americas (Spillane et al. 1987). McPhaden and Taft (1988), using subsurface data collected at three equatorial moorings in the eastern Pacific, found the dominant Kelvin wave period to be near 70 days, the dominant zonal wavelength to be about 13000–14000 km (i.e., about the width of the Pacific basin), and the dominant phase speed to be around 2.4 m s⁻¹, which is similar to that of the gravest baroclinic mode based on the observed stratification in the equatorial Pacific (e.g., Giese and Harrison 1990). These findings were subsequently confirmed by KMW using observations from the denseTropical Atmosphere Ocean (TAO) mooring array (Hayes et al. 1991), which spans the equatorial Pacific from near 150°E to near 90°W. KMW also showed the typical vertical displacement of the thermocline produced by these waves to be about 20 m and that these waves exhibit a distinct seasonal variation, with maximum activity occurring from November through March.

These Kelvin waves are of interest because they are nondispersive and are the fastest propagating waves along the equator: Localized surface forcing in the west can be communicated to the eastern Pacific in less than 3 months. One such mechanism by which these waves can remotely affect the tropical climate is via zonal advection of the mean sea surface temperature (SST) gradient. Intraseasonal SST anomalies of up 0.5°C have been attributed to such advection (e.g., Johnson and McPhaden 1993; KMW). A coupled feedback between the horizontal extent of the surface westerlies/convection over the western Pacific and the strength of the intraseasonal Kelvin waves has been proposed by KMW as a mechanism by which these intraseasonal Kelvin waves can cause a steplike, eastward progression of high SST and surface westerlies across the Pacific in a manner reminiscent of the onset of an El Niño.

The Kelvin waves are thought to be forced by intraseasonally varying zonal stress anomalies across thewestern portion of the Pacific basin: Little coherence between the surface wind and these Kelvin waves in the eastern Pacific has been reported, while strong coherence between surface zonal winds in the western Pacific and Kelvin wave activity in the central and eastern Pacific has been found (e.g., Enfield 1987; KMW). Previous modeling studies have implicated synoptic-scale (i.e., timescales less than about 2 weeks and space scales of about 1000 km) westerly wind bursts in the western Pacific (e.g., Giese and Harrison 1991) as a forcing mechanism of the Kelvin waves. Such wind bursts increase in activity prior to and during El Niño (Luther et al. 1983) and hence are thought to perhaps play a role in the evolution of El Niño. However, the coherence of the surface zonal winds in the western Pacific with the intraseasonal Kelvin wave activity in a broad band centered on 70 days has led to speculation that the Kelvin waves are predominantly forced by the atmospheric Madden–Julian oscillation (MJO: Madden and Julian 1972), which dominates the intraseasonal variability of the troposphere in the western Pacific. For instance, Enfield (1987) suggested that the forcing by the MJO occurs predominantly west of the date line, where its eastward phase speed is relatively low (∼5 m s⁻¹; e.g., Madden and Julian 1972) and amplitude of surface winds is large.

The purpose of the present study is to determine the extent to which the atmospheric MJO is responsible for forcing the observed intraseasonal Kelvin waves. To fully understand how the MJO forces the intraseasonal Kelvin waves requires that the discrepancy between the central frequency of the observed Kelvin waves (∼70 days) and the atmospheric MJO (∼50 days) be explained. KMW offered a tentative explanation by assuming, to first order, that the zonal stress anomalies produced by the MJO in the western Pacific are nonpropagating patches. With such an assumption, KMW showed that the Kelvin wave response to such an imposed patch of stress falls off dramatically as the period of the forcing decreases toward the time required for the Kelvin wave to travel across the patch (i.e., the forcing would then integrate to zero) and as the patch width decreases (see also Tang and Weisberg 1984). For patch widths typical of the surface zonal wind anomalies associated with the MJO, KMW show that the calculated response plummets at just the central frequency of the atmospheric MJO (i.e., 50 days). They then argue that this high-frequency cutoff helps explain the apparent discrepancy between the central frequencies of the oceanic Kelvin waves and the atmosphere by shifting the part of the zonal wind spectrum responded to by the ocean to lower frequencies, which also have larger horizontal scales. However, the surface zonal wind anomalies associated with the MJO are by no means fixed in space. Systematic eastward propagation across the western Pacific occurs at about 3–5 m s⁻¹ (e.g., Hendon and Salby 1994). This eastward propagation would be expected to enhance the Kelvin wave response over thatof a standing patch, as the forcing should be near resonance with the gravest baroclinic Kelvin waves (e.g., Tang and Weisberg 1984). The impact of this near-resonance on the frequency of the response has not been elucidated.

The apparent lack of correlation between the intraseasonal Kelvin wave activity and surface winds in the central and eastern Pacific (e.g., Enfield 1987), despite a well-defined (albeit weaker than in the west) signal of the MJO in the surface winds there (e.g., Hendon and Salby 1994), also needs explanation. The signal of the MJO east of the date line does have, however, a distinctly different character than to the west. West of the date line the surface zonal winds move eastward along with anomalous convection at 3–5 m s⁻¹. East of the date line where the convective signal is diminished, the surface zonal winds move eastward at greater than 10 m s⁻¹. Hendon and Salby (1996) interpreted this differing atmospheric behavior as a forced response west of the date line and a radiating response east of it. The change in phase speed results from the tendency for the radiating response to be preferentially excited by the gravest (largest zonal scales) components of the forcing. The forcing (i.e., the coherent large-scale convective anomaly) is predominantly wavenumber 2 with 50-day period west of the date line, consistent with the observed eastward phase speed of about 5 m s⁻¹ there. The localization of the convective forcing west of the date line implies that a significant component in wavenumber 1 (and higher wavenumbers as well) exists. Hence, east of the date line, where the local forcing is absent and the radiating response dominates, a wavenumber-1 structure is observed in the zonal wind but, because it has the period of the forcing, the eastward phase speed is twice that west of the date line. This increased phase speed east of the date line implies that the zonal winds will not project as efficiently onto the oceanic Kelvin waves as the winds do farther west. In fact, the surface zonal winds in the eastern Pacific may even systematically detract from the waves arriving from the west. The lack of coherence between zonal stress in the Kelvin waves in the central and eastern Pacific thus needs to be reexamined.

In contrast to KMW, who concentrated on the coherence with the zonal winds only in the western Pacific, the manner in which the zonal stress anomalies induced by the MJO across the entire Pacific basin interact with the observed intraseasonal Kelvin waves will be addressed. A linear model of equatorial thermocline displacements will also be forced by an observed multiyear record of intraseasonally varying surface stress. The skill of this model for reproducing the observed spectrum of Kelvin waves, with realistic attributes, will be assessed. Accurate depiction of intraseasonal variability is necessary if studies of the role of these waves in lower-frequency phenomena (e.g., the annual cycle and El Niño) are to be pursued with a simple ocean model such as this.

2. Data and analysis methods

Continuous records of various oceanic and atmospheric parameters are employed for the 7-yr period July 1986 through June 1993. The principal oceanic dataset is the time series of depth of the equatorial 20°C isotherm (Z20), as derived by KMW using subsurface thermal data from the TAO array (Hayes et al. 1991). The equatorial Z20 data are well suited for detecting vertical motion associated with intraseasonal Kelvin waves. Using a variety of gap filling and interpolation strategies, KMW produced a continuous daily time series of Z20 along the equator from 165°E to 110°W on a 5° grid from the subsurface temperatures observed at the TAO moorings. Due to the longitudinal spacing of the moorings and the interpolation scheme employed by KMW, the effective resolution of these data is about 15° longitude and 1 week in time.

Daily averages of ascending and descending narrowband infrared radiance observations, which are converted to broadband estimates of OLR, are available on a 2.5° grid (Gruber and Krueger 1984). Missing values are replaced with temporal and spatial interpolation (Liebmann and Smith 1996). OLR is used to infer areas of deep convection.

To emphasize the large-scale structure of the MJO and the oceanic Kelvin waves, most analyses are performed on data (except Z20) that have been averaged onto a 5° latitude by 10° longitude grid within the domain, 15°N–15°S, 60°E–90°W. To isolate intraseasonal variations, the gridded analyses are filtered (via spectral transforms) to periods 25 < τ < 120 days. This broad range of periods captures the spectral peaks centered near 50 days associated with the MJO (e.g., Salby and Hendon 1994) and centered near 70 days associated with intraseasonal oceanic Kelvin waves (e.g., KMW). Results presented here are not sensitive to the exact width of this filter.

The signal of the intraseasonal Kelvin waves is determined by empirical orthogonal function analysis of the intraseasonally filtered Z20 data. EOF analysis, rather than cross-correlation (or its spectral analog cross- power) is chosen so that no “bull’s eyes” centered onan arbitrarily chosen base point result. The EOF analysis is performed on the covariance (unnormalized) matrix for the entire domain of the Z20 data (165°E–110°W). As will be shown in section 3, a dominant pair of eigenmodes results from the EOF analysis, which depict the eastward propagating Kelvin waves. Coherent variability in other fields, such as zonal stress and OLR, is depicted by multiple linear regression onto the principal components (time series of the eigenvectors) of the two leading eigenmodes.

In order to further demonstrate the association of these waves with the MJO, the predominant signal of the MJO is determined by EOF analysis of intraseasonally filtered OLR over the domain 15°N–15°S, 60°E–90°W. This EOF analysis results in a dominant pair of modes (see also Zhang and Hendon 1997), which describes a convective disturbance propagating eastward at about 5 m s⁻¹ across the Indian and western Pacific Oceans with local period 40–50 days, and is thus taken as the signal of the MJO in convection (see also Weickmann 1983; Lau and Chan 1985; Murakami et al. 1986). As for the analyses based on EOFs of Z20, coherent behavior in other fields, such as Z20 and zonal stress, is found by multiple linear regression onto the leading two principal components of the OLR EOFs.

3. Characteristics of the observed intraseasonal Kelvin waves

The leading pair of modes that emerge from the EOF analysis of Z20 together explain ∼80% of the intraseasonally filtered variance (Table 1). As the principal components are well correlated at a lag of about 18 days (not shown), they depict a zonally propagating disturbance with near 70-day period and hence are taken to capture most of the variability associated with the intraseasonal Kelvin waves.

The coherent zonal stress and OLR with the Kelvin waves is determined by multiple linear regression onto the principal components of the two leading EOFs of Z20. The variance explained by the regressions is then computed in a moving, 140-day window. Figure 1 displays these time-varying explained variances for stress (heavy solid curve) and OLR (dotted curve) over the domain 15°N–15°S, 60°E–165°W (which is where the strongest MJO variability in the atmosphere is confined;e.g., Salby and Hendon 1994). Also shown are the time series of variance of Z20 reconstructed from the leading two EOFs (light solid line) and associated explained variance for Z20 (light dashed curve) along the equator for 165°E–110°W (the domain of the available Z20 data). A distinct seasonal variation, consistent with that found by KMW, is apparent with peak values approaching twice the mean amounts from about October through March. The seasonal variation of explained variance is similar to that of the signal of the eastward propagating MJO in convection (Salby and Hendon 1994). Significant interannual variations in these explained variances are apparent as well, but are not further discussed.

The peak explained variance for the regressed zonal stress reaches about 30%. Regressed OLR exhibits nearly identical behavior (see also Salby and Hendon 1994;Hendon and Glick 1997). Peak explained variance for Z20, on the other hand, reaches upward of 90%. This large explained variance may result partially from the coarseness of the TAO observing network and the spatial and temporal interpolations used by KMW in the construction of the Z20 time series, which both tend toremove high frequencies and small space scales from the Z20 variability.

The structure and propagation of the disturbance captured by the two leading eigenmodes of Z20 is depicted by lag-regression of the individual fields onto the associated principal components (Table 2). Regressed fields are displayed for 1.5 standard deviation anomalies of the two principal components. Figure 2 displays these Z20 (shaded) and zonal stress (contoured) anomalies along the equator. The Z20 field depicts an eastward propagating disturbance at about 2.3 m s⁻¹, with a wavelength of about the width of the Pacific basin and period of about 70 days. These characteristics are consistent with previous analyses of intraseasonal Kelvin wave activity in the equatorial Pacific (McPhaden and Taft 1988; KMW). Hence, the disturbance depicted by the EOF analysis is taken to be that of the dominant intraseasonal Kelvin waves. Maximum amplitude of the Z20 perturbations occurs around 120°–130°W, where some indication of a reduction of phase speed occurs (see also Johnson and McPhaden 1993). The coherent zonal stress exhibits eastward propagation at about 3–5 m s⁻¹ west of the date line, with more rapid eastward propagation east of it. These zonal stress anomalies possess a Gaussian-like structure in latitude with e-folding scale of about 7.5° (not shown). These features are characteristic of the surface zonal wind anomalies produced by the MJO (Hendon and Salby 1994). A similar zonal and meridional structure was also found for the leading complex eigenmode of the intraseasonally filtered 850-mb zonal wind by Enfield (1987).

The connection between the observed intraseasonal Kelvin wave activity and the MJO is further emphasized by considering the OLR anomalies that are coherent with the intraseasonal Kelvin waves (Fig. 3a). For comparison, the OLR anomalies, based on EOF analysis of intraseasonally filtered OLR, are shown in Fig. 3b. The leading two EOFs of intraseasonally filtered OLR, which are taken together to capture the signal of the MJO in convection (e.g., Zhang and Hendon 1997), account for 35% of the intraseasonally filtered OLR variance (Table 1). Also shown in Fig. 3 are the coherent zonal stress anomalies for each case (see also Table 2). Note, the anomalies are shown from 60°E to 90°W in order to emphasize the global extent and zonal propagation of the MJO. For both EOF analyses (Fig. 3), zonal stress and OLR anomalies, with a local zonal wavenumber-2 structure, propagate systematically from the Indian Ocean eastward to about the date line. Westerly zonal stress anomalies lag enhanced convection (reduced OLR) by about 5–10 days. The convective anomalies disappear near the date line, at which point the eastward phase speed of the zonal stress anomalies increases. This behavior, captured in both analyses, is characteristic of the MJO (e.g., Hendon and Salby 1994). Note, however, that the dominant period of the stress and OLR perturbations based on the Z20 EOFs is longer than that typically associated with the MJO (70 days as compared to 40–50 days).

This discrepancy in frequency is highlighted by considering power spectra of zonal stress at 155°E, which is representative for points west of the date line (Fig. 4). The spectrum of raw stress shows a broad spectral peak near 50–60 days, but with elevated power between 30 and 80 days. The regressed stress based on the EOFs of OLR (dotted line in Fig. 4), which is one measure of stress coherent with the MJO, contains only about one-third of the original power, is peaked at about 50 days, and has a long high-frequency tail. In contrast, the spectrum of the stress based on the EOFs of Z20 (long dashed line, Fig. 4) exhibits a spectral peak near 70 days with little power at periods shorter than 50 days. Hence, we conclude that the observed Kelvin waves are coherent with the low-frequency tail of the broad intraseasonal spectrum produced by the MJO: Little of thestress variance at periods less than 50 days appears to be associated with the Kelvin waves.

4. Forcing of the intraseasonal Kelvin waves

Examination of Fig. 2 suggests that the intraseasonal Kelvin waves are predominantly forced by eastward moving stress anomalies west of about 170°W produced by the MJO. These stress anomalies have a local wavelength in the western Pacific of about 15000 km. Positive, but weaker, projection appears to continue eastward to about 130°W. East of here, the Z20 perturbations move into stress anomalies of the opposite sign, due to the increased eastward phase speed of the stress anomalies, relative to the Z20 anomalies. The maximum amplitude of the Z20 anomalies occurs near 130°W as well.

Whether the stress pattern in Fig. 2, which is characteristic of the MJO, indeed is able to produce realistic intraseasonal Kelvin wave activity is now addressed. To do so, the Kelvin wave response to this stress is computed in the context of a reduced gravity or 1½-layer system. This system represents the ocean as two homogeneous layers, where the shallow water phase speed is given by

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Here g is gravity, H is the mean upper-layer thickness, ρ is the density of the upper layer, and Δρ is the density difference between the upper and lower layers. Using the observed phase speed of 2.3 m s⁻¹, and assuming the equatorial Pacific is equivalent to two homogeneous fluids with their interface at the mean depth of Z20 (about 125 m), leads to an estimation of Δρ/ρ = 0.0043. This is somewhat higher than typically assumed for the equatorial Pacific [e.g., Philander (1990) uses 0.002], but not unreasonable (e.g., Kessler and Taft 1987). Previously, Kessler and McPhaden (1995) indicated that this simple reduced gravity formalism is able to accurately depict the observed thermocline perturbations produced by the intraseasonal Kelvin waves despite apparent contribution from at least two baroclinic modes of the continuously stratified system. Kessler and McPhaden argue that, as long as the forcing is not too far removed from the response, which is the case here where positive forcing appears to occur as far east as 130°W, proper choice of the reduced gravity parameters allows close simulation of the variability resulting from multiple vertical modes.

Following Gill and Clarke (1974), the amplitude of the Kelvin wave response is determined by integrating the zonal stress along the characteristic x − ct = const from the western edge of the Pacific basin to an interior point in question. The resulting upper-layer thickness anomaly (e.g., Kessler and McPhaden 1995) is

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Here T(x, t) is the normalized meridional projection ofthe zonal stress τ^x onto the Kelvin mode with shallow water phase speed c:

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The meridional integration is only extended to 30° latitude, which is more than adequate due to the narrow meridional scale of the oceanic Kelvin waves; ρ = 1000 kg m⁻³ and c is set to 2.3 m s⁻¹. The impact and implication of the choice of the shallow water phase speed is considered below. In Eq. (1) x₀ is the distance from the western edge of the basin, at which the amplitude is to be determined. The characteristic starting at x = 0, t = 0 thus arrives at x₀ at time x₀/c.

The ability of this Kelvin wave model to reproduce the observed spectrum of waves is considered by forcing the model with the observed intraseasonally varying zonal stress for the period 1986–93. The amplitude in the 55–100 d band of the predicted thickness anomalies and of the dominant mode of Z20 are shown in Fig. 5a. The coherence squared and phase angle between the predicted thickness anomalies and the dominant mode of observed Z20 are shown in Figs. 5b and 5c. Also shown in each panel of Fig. 5 are the model results based on a range of shallow water phase speeds between 2.7 and 1.7 m s⁻¹, which spans the phase speeds of the first two baroclinic modes based on observed stratification in the equatorial Pacific (e.g., Giese and Harrison 1990).

In general, the results based on a shallow water speed of 2.3 m s⁻¹ appear to give the largest coherence, smallest phase angle error, and most similar amplitude as compared to the observed intraseasonal behavior. Note especially that upward of 80% of the observed Kelvin wave variance in the eastern Pacific, where the amplitude is largest, is captured by the characteristic model. However, compared to the observed amplitude, use of a 2.3 m s⁻¹ phase speed results in an underprediction of the amplitude and a negative phase angle error (i.e., observations lag the prediction) in the eastern Pacific. That a lower phase speed (i.e., 2.1 m s⁻¹) gives larger amplitude and smaller phase angle error in the eastern Pacific suggests that the observed Kelvin waves there may be better represented with a slower shallow water phase speed (see also Kessler and McPhaden 1995), which may result from the zonal variation of the thermal stratification across the Pacific (e.g., Giese and Harrison 1990).

In order to demonstrate how the dominant Kelvin waves are forced in the model, the 7-yr time series of predicted thickness anomalies, based on a 2.3 m s⁻¹ shallow water phase speed, is subjected to EOF analysis. As for the observed Z20, a dominant pair of modes emerges, which together explain 79% of the intraseasonal variance (Table 1). A longitude–time plot of the reconstructed thickness anomalies and associated zonal stress anomalies is shown in Fig. 6. The similarity with the dominant mode from observations (Fig. 2) is striking. In particular, maximum amplitude of the predicted thickness anomalies occurs near 130°W, which agrees favorably with the observations. East of this longitude, the amplitude diminishes slightly, as the 2.3 m s⁻¹ Kelvin wave characteristic moves into stress forcing of the opposite sign. West of this longitude, where the stress forcing is seen to move eastward at a phase speed similar to that of the Kelvin wave, near-resonant forcing (e.g., Tang and Weisberg 1984) is indicated.

The model-predicted thickness anomalies do appear to have slightly higher frequency and weaker amplitude than observed. These differences are highlighted by considering the spectrum of observed Z20 and model- predicted thickness anomalies at 133°W, which is approximately the longitude of largest amplitude. The thinsolid line in Fig. 7 is the spectrum of observed Z20 reconstructed from the leading two EOFs. Recall that about 80% of the intraseasonal variance is captured by these leading two EOFs. Hence, the spectrum of reconstructed Z20 appears nearly identical to the spectrum of the input data, at least in the intraseasonal range of frequencies considered here (see also KMW). The dominant peak is near 70 days, with a sharp drop off at higher frequencies. The spectrum of reconstructed thickness anomalies based on the dominant two EOFs of the model is shown as the thick line. As anticipated from Fig. 5, the predicted amplitude is slightly deficient at lower intraseasonal frequencies.

Examination of the spectrum of the zonal stress that is coherent with the dominant Kelvin waves in the model (heavy solid curve in Fig. 4) shows that the dominant Kelvin mode in the model is, in fact, not responding to stress with periods shorter than about 50 days, despite the bulk of the intraseasonal stress power occurring there. Hence, the characteristic model does show astrong tendency to selectively respond to the lower, but less energetic, intraseasonal frequencies of zonal stress associated with the MJO. However, as compared to the spectrum of stress that is coherent with the observed Z20 anomalies (long dashed line in Fig. 4), the characteristic model does not show enough sensitivity at low frequency. This point is driven home by considering the model response to just the zonal stress that is coherent with the observed Z20 anomalies. The spectrum of this stress (Fig. 4) displays little power at periods shorter than 50 days and is peaked near 70 days, which is the dominant period of the observed Kelvin waves. The spectrum of the response to this stress (long dashed line in Fig. 7) has a similar shape. In particular, little response at periods less than about 55 days results. However, the power at 70 days is still deficient. Interestingly, the model response to the zonal stress that is coherent with the EOFs of OLR (i.e., the predominant stress associated with the MJO, whose spectrum is shown as the dotted line in Fig. 7) exhibits a similar confinement to low intraseasonal frequencies despite a preponderance of stress variance at high intraseasonal frequencies (i.e., periods less than 50 days; dotted line in Fig. 4).

In summary, upward of 80% of the observed intraseasonal Kelvin wave variance in Z20 is captured by a simple Kelvin wave model forced with observed intraseasonally varying zonal stress. The coherent component of the broad intraseasonal stress forcing with the Kelvin waves, both as diagnosed in observations and in the model, is associated with the lower-frequency components of the MJO. In agreement with previous studies (e.g., Enfield 1987; KMW), the predominant forcing occurs in the western Pacific where the eastward phase propagation of the MJO-induced stress anomalies is similar to that of the gravest baroclinic Kelvin waves. Positive forcing does appear to come from as far east as 130°W, which is the longitude of maximum amplitude both in observations and in the model. The model, in agreement with observations, exhibits marked frequency discrimination such that the higher-frequency, yet more energetic, stress anomalies associated with the MJO produce very little Kelvin wave response. The model, however, appears to be somewhat deficient in its response at the dominant observed frequencies (i.e., 70 days). This suggests that either the simple model is not appropriate or that there are systematic errors in the stress estimates based on the ECMWF analyzed winds. These issues are returned to in section 6.

5. Discussion

To understand better how the lower-frequency, but energetically weaker, components of the MJO-induced stress anomalies, with horizontal structure as seen in Figs. 2, 3, and 6, are producing the observed Kelvin wave spectrum, the Kelvin wave response to some idealized forcing is considered. Previously, KMW examined the Kelvin wave response at the eastern edge of apatch of zonal stress, which varied sinusoidally in time but was otherwise fixed in space. They showed that the amplitude of the response increases with the width and period of the patch (such that the Kelvin waves feel the forcing longer) and approaches zero as the period of the patch approaches the time it takes for the Kelvin wave to traverse the patch (i.e., the forcing then integrates to zero).

The response at the eastern edge (x = L) of these patches is computed by substitution into Eq. (1). The response to the square patch is (see also KMW)

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The response to the standing wave is

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And, finally, the response to the propagating wave is

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The response to the propagating wave maximizes when its phase speed (c_τ = ω/k) equals the shallow water phase, or when ωL/π = c. Using L’Hospital’s rule, the limit of A_pw(L, t) as the phase speed of the patch c_τ approaches the shallow water speed is

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Hence, the response remains finite for finite patch width L (in contrast to the formula presented by Tang and Weisberg 1984), with resonance only occurring for infinite patch width. This near-resonant behavior for finite patch width follows from the notion that the Kelvin wave response is proportional to the time the Kelvin wave feels the forcing. For a limited longitudinal extent of the forcing, the maximum time the Kelvin waves can feel the response occurs when the patch moves eastward at just the Kelvin wave speed.

Examples of the variance of the responses to these idealized wind patches are shown in Fig. 8. The variance of the response at L is calculated by integration of the squared amplitude over one period of the forcing:

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The variance of the response is shown as a function of period of the patch. The patch 7000-km wide is typical of the surface zonal wind anomalies produced by the MJO across the western Pacific (e.g., Fig. 3). The patch 2000 km wide is typical of intense synoptic-scale westerly wind bursts, which have been associated with the onset of El Niño (e.g., Harrison and Giese 1988).

KMW discuss in detail the properties of the response for the square patch. The response goes to zero when the period P = L/c, which for the 7000-km wide patch is at 35 days. The sharp drop off of variance near the central period of the MJO (i.e., near 50-day period) has been attributed by KMW as an explanation for the lower frequency of the observed Kelvin wave response as compared to the stress forcing.

The response to the 7000-km eastward-moving wave exhibits near-resonance at 70-day period, which is where c_τ = c, and which also coincides with the observed spectral peak in Z20. The maximum amplitude at 70-day period approaches the amplitude attained for the square patch as its frequency goes to zero. This occurs because the Kelvin waves cannot feel any more forcing than that for the patch moving eastward at the same phase speed as the Kelvin wave, which equates to leaving a stationary patch on indefinitely. On the other hand, as the frequency goes to zero the response to the eastward moving wave decreases and asymptotes to the response to the standing wave. This is because the standing wave is sinusoidal in longitude as well as time, whereas the square patch is constant in longitude but sinusoidal in time. As the frequency approaches zero, the phase speed of the eastward propagating wave approaches zero but still is sinusoidal in longitude. Thus, at zero frequency the response to the eastward moving wave is weaker than that to the square patch because the total forcing is less due to its sinusoidal structure in longitude.

The weakness of response to the eastward-moving patch of width 2000 km emphasizes the need for small patches to be of large amplitude in order to produce significant Kelvin wave responses. Hence, the amplitude of the stress anomaly for the 2000-km patch would have to be an order of magnitude larger than that of the 7000- km patch to produce a similar response. Such large stress anomalies for similar 2000-km, wide stress patches (∼0.2 N m⁻² as compared to ∼0.015 N m⁻² for the typical MJO-induced anomalies found here) were considered by Giese and Harrison (1990).

6. Conclusions

Intraseasonal Kelvin waves, which contribute predominantly to the variability of the equatorial thermocline in the central and eastern Pacific Ocean, were shown to be forced by the zonal stress anomalies associated with the lower-frequency components of the atmospheric MJO (see also KMW). These stress anomalies typically move eastward at less than 5 m s⁻¹ from the Indian Ocean eastward to about the date line and have a local zonal wavelength of about 15000 km. East of the date line, where the convective component of the MJO weakens, the eastward propagation of the zonal stress anomalies increases to greater than 10 m s⁻¹. This change in propagation has a significant impact on the forcing of the Kelvin waves. In the western Pacific, the phase speed of the stress anomalies is only slightly greater than the phase speed of the gravest baroclinic Kelvin waves. Hence, near-resonant forcing (e.g., Tang and Weisberg 1984) results. Despite the larger phase speeds and weaker amplitudes east of the date line, positive projection of the zonal stress anomalies onto the Kelvin waves continues to occur eastward to about 130°W, which is where the thermocline perturbationshave largest amplitude. East of this longitude, the enhanced phase speed of the stress anomalies results in a negative projection onto the Kelvin waves and their amplitudes diminish farther to the east. The reduction in amplitude in the far eastern Pacific may also result from internal dissipation in the absence of strong forcing in the east.

The observed spectral peak near 70-day period results from a combination of the temporal and spatial characteristics of the observed zonal stress across the Pacific basin. The temporal spectra of zonal stress across the western Pacific exhibits a broad spectral peak, centered on about 50 days, associated with the MJO. The large spatial scale of the stress associated with the MJO (i.e., about 7000-km half-wavelength) implies that essentially no Kelvin waves will be excited at periods less than about 50 days (Fig. 8; see also KMW). Hence, the observed Kelvin wave spectrum peaks at a frequency given by the convolution of the observed stress spectrum (Fig. 4) and response spectrum for the relevant large spatial scales of the observed zonal stress, which is fairly flat at periods greater than about 70 days for zonal scales ∼7000 km (i.e., Fig. 8). Hence, the Kelvin wave response peaks at about 70 days. On top of the this, the near-resonance of the response to an eastward propagating patch of 7000 km extent, which is representative of the MJO-induced zonal stress anomalies across the western Pacific, at 70-day period leads to a relatively large response for relatively small forcing. The response to an idealized propagating patch, however, shows only a 13% decrease in amplitude between 70 and 50 days (cf. Fig. 8), while the observed high-frequency cutoff is considerably more pronounced (cf. Fig. 7). The shallow-water model, forced with observed intraseasonally varying stress, does show a more dramatic high-frequency cutoff (Fig. 7), suggesting that the structure of the actual MJO-induced stress anomaly is not perfectly represented by the idealized propagating wave considered in section 5.

The shape of the spectrum of the dominant Kelvin mode in the shallow-water model is similar to that observed, but the power is deficient at the dominant spectral peak near 70 days (Fig. 7) and, for that matter, at all lower intraseasonal frequencies (i.e., periods less than about 60 days). This suggests that either the simple linear model is inadequate or the amplitude of the lower intraseasonal frequency stress is deficient. The most obvious deficiencies in the simple model are inclusion of only a single vertical mode and neglect of shoaling of the thermocline across the equatorial Pacific. Kessler and McPhaden (1995) argue that as long as the waves have not moved too far away from the region of forcing, which appears to be the case in the eastern Pacific, where the MJO forcing extends all the way to 130°W, the reduced gravity model can be tuned such that effects of multiple vertical modes are captured. This tuning may not be adequate if the relative contribution or vertical structure of the modes changes across the basin. Forinstance, the observed change in stratification may lead to a shift in frequency and increase in amplitude of individual vertical modes (Bussalachi and Cane 1988):As the waves propagate into the eastern Pacific, their phase speeds reduce and their meridional scales decrease. Conservation of energy flux requires that their amplitudes increase. The shoaling thermocline also leads to a change in the vertical structure of each mode such that the lower modes (those with slower phase speeds and hence lower frequencies) may contribute more to the thermal variability in the upper ocean (Giese and Harrison 1990).

The possible deficiency of the lower intraseasonal frequency stress, based ECMWF wind analyses, may be revealed by comparison to observations from the TAO moorings, which span the equatorial Pacific. However, ongoing integrations of an ocean general circulation model suggest that deficiencies of the simple reduced gravity model are more likely than systematic deficiencies in the stress. Figure 9 displays the spectrum of the Z20 displacements at 133°W produced in a seven year integration of the Gent and Cane (1989) primitive equation model of the Pacific basin. The current version, asdescribed in more detail by Brady and Gent (1994), has seven active layers that describe the upper 400 m, meridional boundaries at 30° latitude, and realistic eastern and western boundaries. Brady and Gent have shown it can realistically simulate the annual cycle of the circulation in the Pacific basin when forced by monthly climatological surface stress and heat fluxes. Here, we have added the observed intraseasonally varying zonal stress for the period 1986–93 to their climatological stress and integrated the model for an additional seven years. The resulting spectrum of equatorial thermocline variability (Fig. 9) agrees more favorably with the observed spectrum of Z20 than does the simple reduced gravity model (Fig. 7), especially with regard to the spectral peak at 70 days. Also shown in Fig. 9 is the coherence squared with observed Z20. Strong coherence (i.e., greater than about 0.5) is observed across the entire intraseasonal frequency band of interest here, suggesting that this multi-vertical-mode model, which produces a realistically sloping and annually varying thermocline, is able to produce a realistic spectrum of Kelvin waves with proper lower intraseasonal frequency amplitude. The mean coherence squared across the 50–85 day intraseasonal band is, however, slightly less than that for the simple reduced gravity model (i.e., Fig. 5b). Such a reduction in coherence may possibly reflect realistic variability generated in the model, which is not coherent with the Kelvin waves, but which is also not resolved in the observed Z20 analyses due to limitations imposed by the temporal and spatial distribution of the mooring array (KMW).

The near-resonant response in the western Pacific suggests that the relatively weak zonal stress anomalies (amplitude ∼0.015 Nm⁻²) associated with the lower- frequency components of the MJO are able to account for the observed intraseasonal Kelvin wave activity. The combination of relatively large longitudinal extent (∼7000 km) and slow eastward phase speed (<5 m s⁻¹) of these stress anomalies act together to produce a relatively large response. It is interesting to note that previous studies have considered the Kelvin wave response to so-called westerly wind bursts (e.g., Harrison and Giese 1988; Giese and Harrison 1990, 1991), whose typical longitudinal extents are ∼2000 km and whose amplitudes are ∼0.2 N m⁻². That the observed seasonally varying Kelvin wave behavior is associated with the weaker, yet broader, MJO-induced stress anomalies suggests that intense synoptic-scale wind bursts, capable of producing a basinwide Kelvin wave response, are relatively rare, but not necessarily unimportant for specific Kelvin wave events (e.g., Lukas et al. 1984; McPhaden et al. 1988). Furthermore, it is also likely that numerous westerly wind bursts, but of modest magnitude, are contained within the MJO (e.g., Nakazawa 1988; Hendon and Liebmann 1994), but that it is their collective longitudinal coherence that gives rise to the observed Kelvin wave behavior. This is not to say that the oceanic response to the individual wind bursts isnegligible. Indeed, Eriksen (1993) has shown that near- resonant gravity waves may be locally excited by relatively rapidly eastward moving (∼10 m s⁻¹) bursts. The large vertically sheared currents produced by these waves may lead to significant mixing and deepening of the mixed layer in the western Pacific, and hence these bursts may play an important role in the climate of the warm pool.