## 1. Introduction

The equatorial Pacific Ocean plays an important dynamic role in the evolution of the global coupled atmosphere–ocean system. In addition to generating strong interannual variations such as the El Niño–Southern Oscillation, this region also exhibits significant intraseasonal variability in the form of the Madden–Julian oscillation (MJO). The MJO is the dominant intraseasonal mode of variability of the tropical atmosphere and has time scales of 30–90 days. Equatorial variations of zonal wind stress associated with the MJO can drive large responses in the current, sea level, and thermocline depth of the upper ocean (e.g., Enfield 1987; Kessler et al. 1995; Ralph et al. 1997). These wind-induced signals can propagate eastward along the equator as Kelvin waves and can cause significant intraseasonal changes in sea surface temperature in the central and eastern Pacific (Zhang and Gottschalck 2002). There is also evidence that Kelvin waves excited by the MJO can play an important role in the onset and development of some El Niño events (e.g., McPhaden 1999; Bergman et al. 2001).

Intraseasonal Kelvin wave signals can be detected in observations of thermocline depth (e.g., Kessler et al. 1995) and sea level observed by satellite altimeters. In addition to Kelvin waves, intraseasonal variations of the equatorial ocean may also be related to dynamic processes related to the propagation of Rossby waves and tropical instability waves. The forcing and propagation mechanisms of these various waves are different. Decomposing the observed variations into components related to each wave type can be helpful in terms of understanding the underlying dynamics and, on a more practical note, developing a forecasting capability. Compared with the observed variations in isothermal depth, the intraseasonal variation of sea level is generally weak. On the other hand, the sea level variations observed by satellite altimeters have much higher spatial resolution than the in situ temperature measurements, and thus permit much finer structures in the ocean response to be defined.

In this study we analyze satellite-based observations of sea level and zonal wind stress from the equatorial Pacific. The first step in the analysis is to extract the MJO-related signal in sea level (*η*_{MJO}) and map the spatial distribution of its variance. The next step is to establish the relationship between *η*_{MJO} and the MJO-related zonal wind stress variability. Previous studies (e.g., Hendon et al. 1998) have used reanalysis winds to represent the time varying surface forcing for intraseasonal Kelvin waves. In this study we use observed surface winds from the multiyear Quick Scatterometer (QuikSCAT) mission. Similar to the analysis of sea level we first extract the MJO-related signal from the zonal wind stress *η*_{MJO}).

To study changes in the propagation of intraseasonal equatorial Kelvin waves, Shinoda et al. (2008) recently conducted sensitivity experiments using an ocean general circulation model and a linear stratified model. They showed that even though the zonal wind variations to the east of the date line are weak, they can increase significantly the speed of the forced Kelvin waves compared with their unforced phase speed. In this study, a much simpler forced Kelvin wave model is used to verify their finding.

## 2. Sea level and wind observations and the MJO index

The sea level data used in this study are gridded Ocean Topography Experiment (TOPEX)/Poseidon altimeter measurements for October 1992–May 2007. They were obtained from the Archiving, Validation, and Interpretation of Satellite Oceanographic data (AVISO) Web site (www.aviso.oceanobs.com). The wind data are gridded QuikSCAT scatterometer measurements at 10 m from July 1999 to May 2007. They were obtained from the Remote Sensing Systems Web site (www.ssmi.com). The bulk aerodynamic formula of Large and Pond (1981) was used to convert wind velocity to wind stress. This study will focus on the equatorial Pacific, where the spatial coverage of both datasets is complete.

The MJO is represented by the bivariate MJO index of Wheeler and Hendon (2004), obtained from the Web site of the Australia Bureau of Meteorology (www.bom.gov.au). This index is calculated from the first two empirical orthogonal functions (EOFs) of multivariate fields of equatorially averaged 850-hPa zonal wind, 200-hPa zonal wind, and outgoing longwave radiation observed by satellite. Projection of the daily observations onto the EOFs, after removal of the annual cycle and interannual variability, yields principal component time series that vary mostly on the intraseasonal time scale of the MJO. The first two principal components are referred to as the real-time multivariate MJO series 1 and 2 and denoted by RMM1 and RMM2, respectively. We note that while the variations of the RMM1 and RMM2 are dominated by MJO signals, they also include contributions from other atmospheric waves, such as the equatorial Rossby waves and the convectively coupled Kelvin waves (Roundy et al. 2009). Further, the use of only the first two principal components may exclude an eastward-extending mode of wind variability (Kessler 2001).

## 3. Relationships between sea level, wind, and the MJO

In this section we first explore the relationship between sea level and the MJO, and then zonal wind stress and the MJO, using statistically based spectral techniques. We then explore the relationship between sea level and wind stress variations associated with the MJO using a simple, physically based model.

### a. Sea level and the MJO

Given the present focus on intraseasonal variability, the sea level observations are first high-pass filtered with a cutoff period of 120 days. The standard deviation of the high-pass-filtered observations is mapped in Fig. 1a for the equatorial Pacific. A high-variance region is clearly evident off the coast of Central America. This is believed to be caused by wind jets passing through mountain gaps (Chelton et al. 2000a,b). In the central equatorial Pacific there is a band of high variance near 6°N that extends from the date line to about 120°W. It is believed that this band is due to tropical instability waves and Rossby waves. There is also a narrow band of elevated sea level variance along the equator that starts at about 170°E and ends near 120°W where it merges with the high-variability band to the north. The elevated variance in this region will be related to intraseasonal Kelvin waves in the following analysis.

A wavenumber–frequency analysis (Wheeler and Kiladis 1999) was performed on sea level observations made in the Pacific within 4° of the equator. Figure 2 shows the logarithm of spectral density as a function of wavenumber and frequency. Dispersion curves of Kelvin and Rossby waves have been superimposed. The phase speeds of the Kelvin wave (1.7 and 2.7 m s^{−1}) span the phase speeds of the first two baroclinic modes calculated from the observed stratification in the equatorial Pacific (e.g., Giese and Harrison 1990). The two dispersion curves for the Rossby waves were calculated using the same equivalent depths as the two Kelvin wave dispersion lines. A clear Kelvin wave signal is evident in the spectral density of sea level at periods centered on about 70 days, consistent with the study of McPhaden and Taft (1988). In the negative wavenumber domain (corresponding to westward propagation), a ridge of elevated spectral density centered on one of the Rossby wave dispersion curves is evident. (Note it was not necessary to remove the background spectrum prior to analysis because the strength of the equatorial waves was sufficiently high that the signals could be identified in the raw spectrum.) Overall, this spectral analysis confirms that most of the intraseasonal sea level variability in the equatorial Pacific is related to Kelvin waves rather than to the equatorial Rossby waves.

*ω*) we calculate the squared multiple coherence [

*κ*

^{2}(

*ω*)] between sea level at each grid point and the two MJO indices. An overall measure of the strength of the relationship between the sea level and the MJO is given by the integrated squared coherence after weighting by spectral density (Oliver and Thompson 2009, manuscript submitted to

*J. Geophys. Res.*):where

*f*(

*ω*) is the power spectral density of sea level. Note that this statistic transforms the coherence, which is a function of frequency, into a frequency-independent value that is constrained to lie between 0 (no linear relationship between sea level and the MJO) and 1 (perfect coherence). The denominator in (1) is the variance of the high-pass-filtered sea level series at a given grid point. The numerator is the variance of that part of the high-pass-filtered sea level data that is perfectly coherent with the MJO. We will henceforth denote the numerator and denominator by

*σ*

_{m}^{2}and

*σ*

_{o}^{2}, respectively.

Maps of *σ _{m}* and

*κ*

*σ*reaches a maximum value of 2.5 cm. The corresponding map of

_{m}*κ*

*σ*

_{o}^{2}) in the central tropical Pacific is coherent with the MJO.

### b. Zonal wind stress and the MJO

The zonal wind stress was analyzed in the same way as the sea level as described in section 3a. Specifically, the zonal wind stresses were first high-pass filtered with a cutoff period of 120 days. The standard deviation of the high-pass-filtered wind (*σ _{o}*) is shown in Fig. 3a. The highest variability is found in the western equatorial Pacific; the variability in the eastern equatorial Pacific is significantly weaker. A wavenumber–frequency analysis (not shown) indicates elevated spectral densities with periods of 30–60 days and positive zonal wavenumbers of 1 to 3. The slope of the ridge of elevated spectral density indicated wind signals that were propagating eastward at a speed of about 5 m s

^{−1}, consistent with the typical propagation speed of the MJO in the western Pacific. The spectral peak is in the same range of frequencies and zonal wavenumbers where most of the MJO energy in convection and tropospheric winds is concentrated. Overall, the spectral analysis provides convincing evidence for eastward-propagating signals related to the MJO in the QuikSCAT zonal wind stress.

Maps of *σ _{m}* and

*κ*

*σ*is also evident in the eastern basin just north of the equator, consistent with the study by Zhang and Dong (2004) of the MJO signal in this region in the boreal summer. The highest values of

_{m}*κ*

*κ*

### c. Changes in sea level due to the MJO

We now ask how much of the sea level variability in the equatorial Pacific can be explained by changes in zonal wind stress associated with the MJO. The first step in our analysis is to extract the time variation of sea level or wind stress that is coherent with the MJO. This was achieved by first calculating the Fourier transform of the variable of interest and multiplying each complex Fourier amplitude by the frequency-dependent transfer functions relating the variable to the two MJO indices (RMM1 and RMM2). Applying the inverse Fourier transform then gave time series of sea level and wind stress (*η*_{MJO} and

*x*−

*ct*equal to a constant) from the western edge of the equatorial Pacific basin to an arbitrary interior point at position

*x*. According to this simple model, the sea level response associated with Kelvin waves is given bywhere

_{o}*ρ*is the density of the upper layer,

*g*is acceleration due to gravity,

*h*is the time mean thickness of the upper layer, and

*c*is the phase speed of the first mode of baroclinic Kelvin wave. Following Hendon et al. (1998),

*T*(

*x*,

*t*) is the normalized meridional projection of the zonal wind stress onto the Kelvin mode:where

*β*is the meridional gradient of the Coriolis parameter. Limits of the meridional integration (Δ) are 30° of latitude, which is more than adequate given the narrow meridional scale of the Kelvin waves. We set

*h*= 125 m and

*c*= 2.7 m s

^{−1}.

Figure 5 shows the standard deviation of *η*_{MJO} at the equator (*σ _{m}*) and sea level calculated according to the above model, both plotted as a function of longitude. The model clearly overestimates

*σ*by a factor of 2 between 150° and 100°W. The model response increases eastward from 170°E to 140°W, and then decreases from 140° to 100°W. This shows that the zonal wind east of 140°W acts to reduce the amplitude of the Kelvin waves. In a similar study, Hendon et al. (1998) are able to obtain realistic variations of the equatorial 20°C isotherm forced with the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis winds. The overestimation in our solution is related to the use of the QuikSCAT winds, which are consistently stronger than the reanalysis products (Gille 2005). To obtain realistic sea level variations we add a damping term with an

_{m}*e*-folding time of 25 days to the linear wave model. The effect on the standard deviation of sea level was dramatic and brought the model into much closer agreement with observations (cf. the solid line and line with open circles in Fig. 5).

This simple model can also be used to assess the relative contributions of winds from different regions through their effect on the Kelvin waves. The thinner line in Fig. 5 shows the solution with damping included but the zonal wind to the east of the date line set to zero. This solution significantly underestimates the observed standard deviation in the eastern Pacific. This shows that

Recently, Shinoda et al. (2008) investigated changes in the propagation speed of intraseasonal Kelvin waves. Based on sensitivity studies using an ocean general circulation model, and a linear stratified model, they showed that zonal wind variations in the central and eastern Pacific can increase the propagation speed of the Kelvin waves. We found similar results using the simple linear model, (2). The changes in the propagation speed of the Kelvin waves, from 3.0 to 3.6 m s^{−1}, are obvious in both the high-pass-filtered sea level and *η*_{MJO} (Figs. 4a,b). The model solution (including the damping term) is shown in Fig. 4c. The model prediction agrees well with observations, and in particular the changes in propagation speed are well reproduced. Figure 4d also shows the model solution but with the wind forcing set to zero east of the date line. In addition to a decrease in sea level variance, the wave propagation speeds in the central and eastern Pacific are now constant (and equal to 2.7 m s^{−1}—the assumed phase speed of freely propagating waves). Thus, the results of this simple linear model confirm that the phase speed of the Kelvin wave can be changed significantly by its linear response to wind stress in the central and eastern basins.

## 4. Conclusions

High-pass filtering was used to isolate intraseasonal variations in both sea level and zonal wind stress in the tropical Pacific. The cutoff of the filter was 120 days. The most energetic sea level variations were found in two zonal strips, west of about 100°W and centered on the equator and 6°N, and also in the region off the west cost of Central America. The most energetic zonal wind stress variations were found in the western part of the equatorial Pacific. Spectral analysis reveals that the sea level variations coherent with the MJO (*η*_{MJO}) are confined to a narrow strip along the equator from 150°E to 110°W, where the MJO can account for 50% of the intraseasonal variance. The zonal wind stress variations coherent with the MJO

Using the wavenumber–frequency analysis techniques of Wheeler and Kiladis (1999), it was possible to demonstrate the eastward propagation of intraseasonal Kelvin waves in the sea level observations. To model the generation, propagation, and decay of the Kelvin waves, we used a linear first-order wave equation forced by *η*_{MJO} and highlights the effect of winds east of the date line on the apparent speed of Kelvin wave propagation. The results from our simple model complement those of the more complicated models presented recently by Shinoda et al. (2008).

However, it is worth pointing out that besides winds the phase speeds of Kelvin waves can be influenced by other processes, including nonlinear wave–current interaction and air–sea interaction (Roundy and Kiladis 2006) that cannot be included in this simple model. For example, during July 1997 the positive intraseasonal sea level anomalies propagated eastward at about 1.7 m s^{−1}, much slower than the phase speed of the first baroclinic mode. Roundy and Kiladis (2006) speculated that these slower waves are related to the intraseasonal air–sea interaction. These slower waves cannot be simulated with the ocean-only models (Shinoda et al. 2008) nor with our simple wave model because of the lack of the coupling mechanisms of ocean and atmospheric convection. We note also that the strong damping (at time scales of about 25 days, about half of the time it takes for the Kelvin waves to cross the basin) included in the simple wave model is related to the QuikSCAT winds used in this study, because such a damping term was not needed in the previous study by Hendon et al. (1998) using the atmospheric reanalysis winds. We should certainly be cautious about whether the QuikSCAT product overestimates the actual wind strength, or the strong damping may merely compensate the processes not included in this simple model.

In this study we have focused on linear Kelvin wave dynamics. While our analysis has shown that most of the variation of *η*_{MJO} can be explained by its linear response to *τ*_{MJO}, much interest has been focused recently on the rectification of intraseasonal forcing in the ocean. For example, Roundy and Kiladis (2006) found that coupling between convection and sea surface temperature can translate the intraseasonal MJO to a lower-frequency Kelvin wave during El Niño development. Clearly, nonlinear mechanisms need to be included in models in order to understand the interactions between the MJO-related variations and those occurring at longer time scales.

## Acknowledgments

We thank Hai Lin, Matthew Wheeler, and Eric Oliver for illuminating discussion and helping with analysis techniques, and Paul Roundy and an anonymous reviewer for helpful comments that helped to avoid making overstatements about the simple nature of the results. XZ’s visit at Dalhousie University is supported by the China Scholarship Council and Nanjing University. KRT acknowledges support from NSERC and the GOAPP research network funded by the Canadian Foundation for Climate and Atmospheric Sciences.

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