Observed Structure of Convectively Coupled Waves as a Function of Equivalent Depth: Kelvin Waves and the Madden Julian Oscillation
Paul E. Roundy^{1}
University at Albany
State University of New York
Abstract
The view that convectively coupled Kelvin waves and the Madden Julian oscillation are distinct modes is tested by regressing data from the Climate Forecast System Reanalysis against satellite outgoing longwave radiation data filtered for particular zonal wave numbers and frequencies by wavelet analysis. Results confirm that nearly dry Kelvin waves have horizontal structures consistent with their equatorial beta plane shallow water theory counterparts, with westerly winds collocated with the lower tropospheric ridge, while the MJO and signals along Kelvin wave dispersion curves at low shallow water model equivalent depths are characterized by geopotential troughs extending westward from the region of lower tropospheric easterly wind anomalies through the region of lower tropospheric westerly winds collocated with deep convection. Results show that as equivalent depth decreases from that of the dry waves (concomitant with intensification of the associated convection), the ridge in the westerlies and the trough in the easterlies shift westward. The analysis therefore demonstrates a continuous field of intermediate structures between the two extremes, suggesting that Kelvin waves and the MJO are not dynamically distinct modes. Instead, signals consistent with Kelvin waves become more consistent with the MJO as the associated convection intensifies. This result depends little on zonal scale. Further analysis also shows how activity in synoptic scale Kelvin waves characterized by particular phase speeds evolves with the planetary scale MJO.

Introduction
The tropical atmosphere organizes moist deep convection over a broad range of spatial and temporal scales. The MaddanJulian oscillation (MJO) dominates variability in convection on intraseasonal timescales of roughly 30100 days (Madden and Julian 1994; Zhang 2005). Rainfall associated with the local active convective phase of the MJO (hereafter, active MJO) is in turn organized into smaller scale wave modes and mesoscale convective systems. Convectively coupled Kelvin waves are widely recognized as a leading signal among the population of modes that comprise the sub scale anatomy of the MJO. These waves produce the highest amplitude signals in outgoing longwave radiation (OLR) data near the equator (Wheeler and Kiladis 1999 (hereafter WK99); Straub and Kiladis 2002; Roundy 2008). MacRitchie and Roundy (2012) showed that roughly 62% of rainfall that occurs in the negative OLR anomalies of the MJO between 10N and 10S over the IndoPacific warm pool regions occurs within the negative OLR anomalies of the Kelvin wave band (after excluding those negative anomalies that do not enclose signals less than 0.75 standard deviation). That result represents nearly twice the average rainfall rate per unit area outside of the Kelvin waves but still within the active MJO. MacRitchie and Roundy also showed that potential vorticity (PV) accumulates in the lower to middle troposphere in wakes along and behind the Kelvin wave convection on its poleward sides, and that this PV remains in the environment for longer than the period of the Kelvin waves. The enhanced PV spreads pole ward behind the waves, and it becomes part of the rotational structure of the MJO itself. Another portion of the rotational response to convection coupled to Kelvin waves propagates eastward with the waves, yielding lowlevel cyclones poleward of the equatorial convection (Roundy 2008). The response to deep convection moving eastward with convectively coupled Kelvin waves makes them similar in many respects to the geographically larger MJO. On the other hand, these patterns distinguish observed convectively coupled Kelvin waves from theoretical Kelvin waves of Matsuno (1966) and Lindzen (1967), which do not include meridional circulation. Nevertheless, many authors acknowledge that Kelvin wave dynamics dominate their evolution because of their dispersion characteristics and because of the relationship between wind and pressure observed in the lower stratosphere away from the deep convection, which consistently shows westerly wind in the ridge and easterly wind in the trough, with little meridional circulation. Although the MJO clearly modulates Kelvin wave activity, amplitudes, and propagation speeds (Straub and Kiladis 2003; Roundy 2008), these waves occur independent of the MJO.
Although several authors during the 1980s and 1990s suggested that the MJO itself might be a modified moist Kelvin mode (e.g., Lau and Peng 1987; Wang 1988; Cho et al. 1994), the idea has since fallen out of favor for several reasons. First, the relationship between zonal wind and pressure anomalies in the MJO appears to be reversed or dramatically offset from that of Kelvin waves, with westerly wind anomalies frequently appearing in the pressure trough collocated with the deep convection (e.g., Madden and Julian 1994; Zhang 2005). Second, a spectral peak associated with convectively coupled Kelvin waves appears to be distinct from that of the MJO (Kiladis et al. 2009), suggesting that the two have phase speed distributions that might not overlap. Third, zonal wave number frequency spectra of OLR data suggest that the spectral peak of the MJO extends across a broader range of wave numbers at a given frequency than does the spectral peak associated with the Kelvin waves, giving the impression of a flat dispersion relationship, even though most of that signature can be explained by geographical variation in MJO propagation rather than true dispersion. This perspective is supported by composite MJO events plotted in the longitudetime lag domain (such as by Hendon and Salby 1994), which show structures favoring wave number 2 over the warm pool (consistent with opposite signed anomalies of convection over the Indian and western Pacific basins) and a half wave number 1 across the western hemisphere. Such half wave number 1 signals project more onto wave number 1 than any other wave number, as shown by a simple application of the Fourier transform in space and time to a perfect eastwardpropagating wave number 1 sine wave that is set to zero in one hemisphere and left alone in the other (a synthetic half wave number 1 pattern). Such geographical variations in MJO propagation must project onto different portions of the spectrum. Seasonal variations in MJO propagation must also project onto different portions of the spectrum. A global wave numberfrequency spectrum analysis conglomerates all of these varying signals together, such that a spectral peak aligned in a particular pattern does not necessarily imply wave dispersion.
A more careful look at each of these characteristics casts some doubt on the assertion that the MJO and Kelvin waves are distinct. First, the algorithm of WK99 would artificially enhance the extent of the spectral gap between the MJO and Kelvin peaks. WK99 normalized their OLR spectra by dividing by a smoothed background spectrum. This background spectrum was obtained by smoothing the original spectrum by an arbitrary number of repeated applications of a 121 filter in frequency and in wave number. This approach conserves the total power in the spectrum but redistributes power in the MJO peak into its surrounding neighborhood, including the region of the spectral gap. This artificial increase in background power would reduce the normalized power there, making the MJO and Kelvin peaks appear better separated. For reference, Fig. 1 shows a wave number frequency spectrum of OLR calculated in a similar manner. The more objective spectrum analysis of Hendon and Wheeler (2007) confirms the presence of a local minimum in power in the spectrum, but not as pronounced as suggested by WK99.
Recently, Roundy (2012) integrated wavelet power in the zonal wave number frequency domain over geographical regions where the 100day low pass filtered 850 hPa zonal winds are easterly or westerly. He found that the gap in the global OLR spectrum derives entirely from regions of easterly lowlevel background flow. The spectrum integrated over regions lowlevel westerly winds has power decline smoothly from the maximum in the MJO band, with no evident spectral gap. Thus the source of the spectral gap is not over warm pool zones where MJO convective signals attain their greatest amplitude. The lowest rate of decline of power occurs along Kelvin wave dispersion solutions between equivalent depths of 5 and 8m. This result suggests that Kelvin waves also propagate eastward more slowly over the warm pool than over trade wind zones. Signals in trade wind zones dominate the global spectrum because these zones occur over more of the global tropics for more of the time than do signals in warm pool westerly wind zones, even though the individual events over the warm pool zones average higher in amplitude.
Equatorial beta plane theories suggest that Kelvin waves are non dispersive except at the shortest wavelengths (e.g., Roundy and Janiga 2012), but variation in coupling between the waves and deep convection apparently results in a large range of phase speeds across the full population of events (Roundy 2008). Although many of these Kelvin waves propagate eastward at 1517 ms^{1} around much of the globe, or roughly twice the phase speed of the MJO, Roundy (2008 and 2012) showed that they tend to propagate more slowly over the warm pool zones. He also showed that synoptic scale Kelvin waves propagate eastward even more slowly as they move through the local active convective phase of the MJO. The same Kelvin wave disturbance can circumnavigate the entire globe, with its phase speed changing continuously with the amplitude of the associated convective signal. This observed variation in phase speeds leaves open the possibility that long Kelvin waves and the MJO may have overlapping dynamics because their phase speed distributions might overlap. These results demonstrate that the spectral characteristics of the MJO and the Kelvin waves are not as distinguishable as previously thought.
The relationship between zonal wind and pressure remains a factor whereby Kelvin waves and the MJO might be distinguishable. Since the pressure wind relationship differs between dry Kelvin waves and the observed MJO, and since the observed distribution of frequencies associated with Kelvin waves are higher than the comparable distribution for the MJO, the pressure wind relationship associated with eastwardmoving signals in OLR data must vary with frequency. If the prevailing view that Kelvin waves and the MJO are distinct modes is correct, the presence of both modes would yield a particular pattern of transition in frequency between the spatial patterns associated with Kelvin waves and those associated with the MJO. At low wave number, the spectral peaks of Kelvin waves and the MJO are proximate to each other. Since the spectral characteristics of both the MJO and Kelvin waves vary substantially from event to event, proximity of the two peaks suggests that there must be overlap between the spectra of the two phenomena. If the MJO and Kelvin waves represent distinct modes in which the pressurewind relationship is not a function of frequency (consistent with the prevailing view), then at some point in spectrum between the peaks of the two modes, both signals would be present and explain roughly the same amount of variance in geopotential height anomalies. Both modes would have lowlevel westerly wind anomalies collocated with negative OLR anomalies, but the associated geopotential height anomalies are strongly offset or opposite. Thus, active convective anomalies at that frequency would be associated with negative geopotential height anomalies with one mode and positive with the other mode. Statistical analysis to extract the average coherent pattern associated with the active convection at that frequency without distinguishing between the modes would thus yield significant lower tropospheric westerly wind anomalies associated with active convection but no significant geopotential anomalies because the two opposite signals would wash each other out. If, however, only one dominant coherent mode exists, with structure modulated by the intensity of the associated rainfall, then the phase relationship between wind and geopotential might shift as a continuous function of frequency, with no geopotential amplitude minimum associated with signals at frequencies between the two extremes.
Statistical analysis of observations and reanalysis data might shed light on the nature of the transition of spatial structures as a function of frequency between the spectral peaks that we associate with Kelvin waves and the MJO. Recently, Roundy and Janiga (2012) applied zonal wave numberfrequency wavelet analysis and simple linear regression to assess the structure of convectively coupled mixed Rossby gravity (MRG) waves characterized by specific zonal wave numbers and frequencies. They applied wavelet analysis at a particular zonal wave number and a specified frequency to generate a time index of the corresponding signals. Regression of fields of data against that index reveals the spacetime structures of the patterns corresponding to those signals. By choosing frequencies consistent with a selected equivalent depth at several different individual wave numbers, they showed how MRG wave structures vary with wave number along particular shallow water model dispersion curves. A similar analysis of signals proximate to the Kelvin wave peak in the OLR spectrum might suggest how Kelvin wave structures change with equivalent depth (h), or how structures of coherent disturbances change between the Kelvin and MJO spectral peaks. The purpose of this work is to apply this technique to better understand what observations suggest about the extent to which the MJO and long convectively coupled Kelvin waves can be considered independent phenomena and to enhance our understanding of interactions between short Kelvin waves and the MJO.
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