convectively coupled equatorial waves; wavelet analysis; non-traditional Coriolis terms

Transcription

1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 138: , April 212 B Analysis of vertically propagating convectively coupled equatorial waves using observations and a non-hydrostatic Boussinesq model on the equatorial beta-plane Paul E. Roundy* and Matthew A. Janiga Department of Atmospheric and Environmental Sciences, University At Albany, State University of New York, Albany, NY 12222, USA *Correspondence to: P. E. Roundy, Department of Atmospheric and Environmental Sciences, DAES-ES351, Albany, NY 12222, USA. roundy@atmos.albany.edu A non-hydrostatic model on the equatorial beta-plane is solved for the most basic solutions of vertically and zonally propagating internal plane waves. Solutions tilt in the vertical and propagate upward, with buoyancy as a principal restoring force. Results indicate that when frequency is of the same order of magnitude as a reduced buoyancy frequency, the shallow-water model equivalent depth depends on frequency. One consequence of this dependence is that Kelvin waves become dispersive at high frequencies. In a complementary observational analysis, linear regression and a space-time wavelet spectrum analysis of observed convectively coupled mixed Rossby gravity (MRG) waves are applied to estimate vertical wavelengths that are consistent with the strongest signals associated with observed convectively coupled waves at specific zonal wave numbers and frequencies. Substitution of these dispersion parameters into the model yields theoretical structures characterized by vertical and horizontal wavenumbers and frequencies similar to those observed in the lower and middle troposphere. These model solutions demonstrate that the Coriolis terms associated with the horizontal component of the earth s angular momentum explain substantial meridional tilts and phase shifts between quantities associated with observed convectively coupled waves, especially proximate to the surface of the Earth. Copyright c 211 Royal Meteorological Society Key Words: convectively coupled equatorial waves; wavelet analysis; non-traditional Coriolis terms Received 18 January 211; Revised 8 September 211; Accepted 7 November 211; Published online in Wiley Online Library 12 December 211 Citation: Roundy PE, Janiga MA Analysis of vertically propagating convectively coupled equatorial waves using observations and a non-hydrostatic Boussinesq model on the equatorial beta-plane. Q. J. R. Meteorol. Soc. 138: DOI:1.12/qj Introduction Convectively coupled equatorial waves interact with and organize moist deep convection in the tropics. They influence tropical cyclogenesis (e.g. Dickinson and Molinari, 22; Bessafi and Wheeler, 26; Frank and Roundy, 26) and interact with the Madden Julian Oscillation (MJO, e.g. Zhang, 25; Roundy, 28). Many authors have drawn analogues between these waves and zonally propagating wave solutions of the shallow-water model on the equatorial beta-plane (Matsuno, 1966; Lindzen, 1967; Kiladis et al., 29, and references therein). These solutions provide horizontal structures and dispersion characteristics that are remarkably similar to many observed waves, given the crude approximations made. In spite of the successes of these shallow-water models in predicting the basic patterns of horizontal structure and dispersion, they cannot directly represent many aspects Copyright c 211 Royal Meteorological Society

2 Analysis of Vertically Propagating Convectively Coupled Equatorial Waves 15 of the observed waves without substantial modification (e.g. Yang et al., 27a, 27b, 27c; Roundy, 28). For example, wave structures tend to tilt meridionally against the direction of propagation with increasing latitude (e.g. Kiladis et al., 29, their figure 12). Many observed waves would rely on buoyancy as a principal restoring force in an atmosphere statically stable to moist deep convection on the spatial scales of the waves, (e.g. Kiladis et al., 29, and references therein). Lindzen (1974), Holton (24), Kiladis et al.(29) and others have overcome this limitation by modifying the shallow-water model to include vertical variation of density. If a rigid lid is not applied at the top of a model troposphere and the atmosphere is assumed to be stably stratified on the spatial scales of the waves, solutions include wave structures that tilt in the vertical, with dispersion characteristics and horizontal structures that depend on vertical wavenumber and the buoyancy frequency (Lindzen, 23). The horizontal structures and dispersion properties of waves in these models can also be obtained from a shallow-water model by choosing a particular equivalent depth. This paper investigates how the vertical structure of convectively coupled waves changes with the zonal scales of the waves, through analysis of theory and observations. This work also provides a convenient opportunity to analyse through a combination of simple theory and observations the influence of the complete Coriolis force on a subset of observed convectively coupled waves. Recently, Kasahara (23a, 23b) postulated that the horizontal component of the Earth s angular velocity would be relevant to the dynamics of non-hydrostatic, diabatically driven circulations in the tropics, through the cosine Coriolis terms in the momentum equations (see Gerkema et al.(25) for a review of these terms). For simplicity, we label these terms the non-traditional Coriolis terms. These terms couple zonal and vertical flow, and would be balanced in the vertical by buoyancy in an atmosphere statically stable to deep convection on the spatial scales of planetary to synoptic scale waves. Recent works have demonstrated that the non-traditional Coriolis terms play important roles in the dynamics of deep atmospheres such as that of Jupiter (e.g. Yano, 1994, 1998, 22; Yano et al., 23, 25). Due to a relatively weak density stratification of the deep oceans, others have recently suggested they could be equally important in the equatorial oceans. For example, Fruman et al.(29) discuss their role in the generation of equatorial jet currents. Even in the Earth s atmosphere, which is neither deep nor weakly stratified, the possibility that the non-traditional Coriolis terms play critical roles has not been excluded. This possibility was investigated recently by Kasahara (23a, 23b, 27, 29). Due to a drastic reduction of effective stratification by diabatic heating associated with moist deep convection and maximization of the cosine of latitude near the Equator, the non-traditional Coriolis component might be important in the tropical atmosphere. Recently, Fruman (29) investigated equatorial wave modes in a simple Boussinesq model on the equatorial betaplane that included these terms. He found that these terms do not influence the dispersion properties of equatorial waves, but that they do influence horizontal structures of waves when the ratio /N is large (where is the angular velocity of the Earth, and N is the buoyancy frequency). This modification of horizontal structure is expressed as a meridional tilt imposed on the solutions. Fruman s model is hydrostatic, suggesting that his result is useful only with respect to low frequency motions. However, no previous works have attempted to diagnose the magnitude of the contributions of these terms to any type of observed convectively coupled waves. The present paper analyses the influence of these terms on observed vertically propagating convectively coupled waves in observations, in the context of a simple model. We first discuss a non-hydrostatic model on the equatorial beta-plane similar to the Boussinesq model of Kasahara (23a, 23b), and the simplest forms of its solutions. We apply this model to generalize the analysis of Fruman (29) to include high frequency waves, and we briefly summarize the consequences of our solutions for such waves. We further generalize the result of Fruman (29) to include simple vertical structure. We then apply this simple generalization to analyse a subset of observed synoptic to planetary scale convectively coupled mixed Rossby gravity waves that tilt in the vertical and propagate upward through the troposphere. Finally, we estimate the magnitude of the contribution of the non-traditional Coriolis terms to such observed waves. 2. The model We begin with the Boussinesq model of Kasahara (23a, 23b). We assume a volume mean reference density ρ that is constant in space and time across a selected layer in the lower to middle troposphere. The local density perturbation, ρ, is allowed to vary in a given layer only in terms multiplied by g (the constant acceleration due to gravity), while ρ is the horizontal mean of the departure from the volume mean density ρ. density (which varies with height within the layer). This model would be applicable to waves characterized by small vertical displacements in the flow near the height of the selected layer, even if their characteristic vertical wavelengths are substantially larger than the thickness of the layer. The perturbation velocity equations (u, v, w ) from a resting basic state are: u βyv + 2 w + 1 p =, t ρ x (1) v t + βyu + 1 p =, ρ y (2) w 2 u + 1 p t ρ z = s, (3) the buoyancy equation is s t + N2 w = (4) and the continuity equation is u x + v y + w =. (5) z Buoyancy is represented as s = g(ρ ρ ) ; and s = gρ /ρ (6) ρ the Brunt Väisällä or buoyancy frequency squared is N 2 = g ρ ρ z. (7)

3 16 P. E. Roundy and M. A. Janiga N is assumed to be constant throughout the selected layer, but it could vary between layers, should other layers be considered. is the angular velocity of the Earth, and β is the meridional gradient of the Coriolis force on the Equator. The standard equatorial beta-plane approximation for 2 sin(φ) (where φ is latitude) is βy. We extend the beta-plane approximation by including the non-traditional Coriolis terms, approximated by 2 cos(φ) 2 ; this approximation generates an error of less than 14% at ±3 latitude. p represents the part of the pressure not in hydrostatic balance with the uniform background density ρ + ρ. Our use of (3) and (4) allows us to analyse high frequency waves that were excluded from the analysis of Fruman (29) (see his section 2) because these equations represent a relaxation of the hydrostatic approximation. We generally assume that the vertical acceleration term is negligible, but we retain it to diagnose a more general solution. We analyse that solution in comparison with that obtained after applying the quasi-hydrostatic assumption (e.g. White and Bromley, 1995) to diagnose the scale of the impacts of this term on the propagation of convectively coupled waves in varying background environments. For simplicity, we assume a background state at rest, although previous works have shown that vertical and meridional shear of the zonal wind (e.g. Xie and Wang, 1996) and the vertically averaged zonal wind influence the structures of observed waves Solutions We assume zonally and vertically propagating internal plane wave solutions of the form (u,v, w, p, s ) = (û, ˆv, ŵ, ρ ˆp, ŝ)exp[i(kx + lz υt)], (8) wherethevariableswithcircumflexesareeachfunctions of y only. k and l are zonal and vertical wavenumbers, respectively; ν represents angular frequency. Since the troposphere is not capped to vertical propagation and since we assume that the troposphere is statically stable to deep convection on the spatial scales of the waves, an infinite number of vertical modes would exist (e.g. Lindzen, 23). We assume that wave planes propagate upward into the layer (thus we assume purely radiative boundary conditions at both the top and bottom of the selected layer). The observational analysis (below) demonstrates that patterns revealed by this simple theory successfully represent previously unexplained patterns in the horizontal structures of convectively coupled waves, suggesting that although crude assumptions are made, the result improves the representation of convectively coupled waves over shallowwater models and models that do not account for the non-traditional Coriolis terms. Substitution of (8) into Eqs (1) (5) yields the following: iνû βyˆv + 2 ŵ + ikˆp = (9) iν ˆv βyû + dˆp dy = (1) iνŵ 2 û + ilˆp = ŝ (11) ikû + dˆv + ilŵ = dy (12) iνŝ + N 2 ŵ = (13) To find a solution to (9) (13) in terms of ˆv analogous to the shallow-water model of Matsuno (1966), we begin by solving (13) for ŝ, substitute the result into (11) iνŵ 2 û + ilˆp = in2 ŵ (14) ν and solve (14) for û û = i 2 ν ( (ν2 N 2 )ŵ + νlˆp). (15) Substitute (15) into (9), (1), and (12), solve the resulting (9) for ŵ, substitute the result into (1) and (12), solve (12) for ˆp and substitute the result into (1) to obtain an equation in ˆv only: d 2 ˆv dy 2 + 4iβyl N ν ( 2 ) dˆv l 2 dy + ν 2 k 2 (N 2 ν 2 ) kβ ν (N2 ν 2 ) N ν 2 2i lβ ˆv + N ν 2 ˆv l 2 β 2 y 2 N ˆv = (16) ν2 Next, apply the transformation ˆv = η(y)exp ( i2 ) Ɣy2, (17) where 2βl Ɣ N ν 2. (18) This transformation is applied because it eliminates the first order derivative and the imaginary part of the zeroth order derivative; it also modifies the term in y 2 and generates an equation in η for which there exist familiar solutions. This transformation is similar to that applied by Fruman (29) and by Gerkema and Shrira (25). The similar transformation of Fruman (29) does not depend on ν because of his assumption of quasi-hydrostatic balance (White and Bromley, 1995). When considered in the context of (8), the complex exponential portion of (17) results in a latitude-dependent, zonal phase shift δ in solutions for v, where δ is the object of the complex exponential in (17) with the i omitted (consistent with the phase shift diagnosed by Fruman, 29). After applying the transformation, (16) reduces to d 2 η dy 2 ( l 2 ν 2 + k 2 (N 2 ν 2 ) + kβ ν (N2 ν 2 ) + y2 β 2 l 2 (N 2 ν 2 ) (N ν 2 ) 2 N ν 2 ) η = (19) Equation (19) has the same form as the final equation for ˆv of Matsuno (1966), and is known as a Weber equation (Polyanin and Zaitsev, 23). This equation has a broad set of solutions, but one particular class of solutions includes the discrete meridional modes familiar from equatorial betaplane shallow-water theory (e.g. Matsuno, 1966). This class of solutions occurs only if ( (N 2 ν 2 ) 1/2 )( k 2 kβν βl + l2 ν 2 ) N 2 ν 2 = 2n + 1, (2)

4 Analysis of Vertically Propagating Convectively Coupled Equatorial Waves 17 Figure 1. Dispersion curves in the zonal wavenumber-frequency domain for solutions to Eq. (24), as expressed in Eq. (24), characterized by the values of N and vertical wavelength given in the figure title. The dashed solution has been deemed unphysical, consistent with Matsuno (1966). Zonal wavenumber is the circumference of the Earth divided by the zonal wavelength. wheren is an integer. Equations (19) and (2) generalize the result of Fruman (29) to include simple vertical structure and a correction in terms of ν that appears due to relaxation of the hydrostatic approximation in this model. n must be a non-negative integer so that (19) can have bounded solutions. Although N 2 = ν 2 is a singular limit, we found no clear evidence of its occurrence in nature away from the surface of the Earth. Discrete solutions to (2) for ν are the numerically calculated roots of the quadratic in k after assuming a range of values for k and specific values of the other parameters. Figure 1 shows the associated dispersion diagram giving the solutions for n = 1 through n = 1, all assuming = s 1 and l 1 = m. Analogous solutions to those of shallow-water theory are labelled on Figure 1, and include the Kelvin, equatorial Rossby (ER), mixed Rossby gravity (MRG), and inertio gravity (IG) including eastward inertio gravity (EIG) and westward inertio gravity (WIG) modes. The dashed curve is associated with a negative root of the Kelvin wave, and it has previously been deemed unphysical because its associated zonal wind anomalies do not taper to zero with latitude, thus making it inconsistent with the equatorial beta-plane assumption (Matsuno, 1966). Although the values of l and N applied to obtain Figure 1 result in dispersion curves that are roughly consistent with the shallow-water equivalent depths of observed convectively coupled equatorial waves (e.g. Liebmann and Hendon, 199; Wheeler and Kiladis, 1999; Roundy and Frank, 24; Yang et al., 27a, 27b, 27c), substantially different values of N and l also yield reasonable dispersion solutions. Comparison of this dispersion relation with the shallow-water model of Matsuno (1966) reveals that the shallow-water model equivalent depth h e = N2 ν 2 l 2. (21) g This result is equivalent to that of Fruman (29) except for inclusion of ν 2. This inclusion indicates that relaxation of the hydrostatic approximation yields dependence of h e on frequency. This dependence would become important when ν 2 is of the same order of magnitude as N 2. If we let β 2 l 2 ((N 2 ν 2 ) (N ν 2 ) 2 = K2 >, (22) then the corresponding solutions for the meridional structure of v anomalies are of the form η(y) = H n ( ) Ky)exp ( Ky2, (23) 2 where H n is the nth Hermite polynomial. Therefore, solutions for ˆv may be obtained from η by applying (17). Setting explicit appearances of equal to (i.e., retaining only the appearances of that are implicit in β) eliminates the contributions of the non-traditional Coriolis terms and reduces the result to the Boussinesq model analogue to the shallow-water model of Matsuno (1966). We assume that a Boussinesq plane wave solution would approximate the structure of some convectively coupled atmospheric waves propagating upward through a layer of the lower to mid-troposphere, from below. In the context of the MRG wave, we assume a wave source in the upper mid-troposphere (such as, perhaps, a tropical cyclone) from which wave energy disperses downward and eastward across the mid- to lower troposphere and upward and eastward across the upper troposphere and the stratosphere. We assume pure radiative boundary conditions at the bottom and top of the layer Kelvin wave Solutions for the Kelvin wave are found by setting v = everywhere and for all time in Eqs (1) (5), and by solving a selection of four of the resulting equations for û. Since four equations can be selected from the original five, we found that two forms of the same solution exist in terms of different combinations of parameters. The same gamma transformation applied above for v also applies to the Kelvin wave solution for, and one form of the solutions for the corresponding η is identical to (23), except that u (the maximum value of the equatorial zonal wind) replaces H n. The dispersion relationship for the Kelvin wave can be found from the selection of four equations noted above, or by substituting 1 forn into (2) (although the corresponding solution forms cannot be obtained directly from (19)). Of the three resulting roots for ν, the positive root of ( ) ν 2 = N2 l 2 k k2 l 2 (24) is the non-hydrostatic analog to the shallow-water Kelvin mode. Equation (24) implies that the Kelvin wave is dispersive at wave numbers much higher than those plotted in Figure 1. This dispersion is a consequence of relaxation of the hydrostatic approximation, and it is relevant only at a high zonal wave number. Wheeler and Kiladis (1999) and many others have demonstrated that observed convectively coupled Kelvin waves exhibit some dispersion especially between wave numbers 5 and 1, but Eq. (24) demonstrates that relaxing the hydrostatic approximation cannot explain this observed behaviour.

5 18 P. E. Roundy and M. A. Janiga 2.3. Reduced moist stability Values for N 2 required to match the dispersion curves to those of observed convectively coupled waves are much smaller than observed values of N 2. Previous works have suggested that the actual value of N 2 felt by a wave is reduced by the influence of the release of latent heat in deep convection coupled to the wave. These works suggest that an effective N 2 can be obtained from the actual N 2 as N 2 = (1 α)n 2 (25) (e.g. Emanuel, 1987; Neelin and Held, 1987; Yano and Emanuel, 1991; Emanuel et al., 1994; Kiladis et al., 29). Here, α is considered a property of the environment in which vertical velocity is positively correlated with the release of latent heat, reducing the effective static stability under vertical motion imposed by a wave. Although some waves might evolve differently, the focus of this work is on waves that propagate in an atmosphere statically stable to moist deep convection on the spatial scales of the waves themselves, with unstable conditions possible on local scales. Such waves would induce wave-scale envelopes of enhanced and suppressed convection without wave-scale vertical overturning circulations. Our use of α under these assumptions is consistent with its application by Emanuel et al.(1994). Although this parameterization is a simplification of the observed system, it has some observational support (Haertel and Kiladis, 24; Haertel et al., 28) and allows for simple analytical solutions to the model that might offer insight into the behaviours of some observed waves. In order to obtain real-valued dispersion parameters from Eq. (2), it is necessary to assume ( ν ) 2 <α<1 < 1. (26) N We thus focus our analysis on a particular subset of atmospheric waves for which α is everywhere <1. Consistent with this view, previous works have suggested that α is close to, but <1 for most convectively coupled waves (e.g. Gill, 1982; Kiladis et al., 29). Observed values of N 2 are substantially higher in the stratosphere than in the troposphere. Parameterized latent heating effectively reduces N 2 in the troposphere through α, which further increases this difference. Hereafter, we focus only on waves of the lower to middle troposphere that tilt in the vertical against the direction of propagation. This distinction of the class of waves that are the focus of this work is important in light of the suggestion by Yano (27) that α might vary substantially in space and time and occasionally even take on values >1 in association with some disturbances. This condition would lead to imaginary values of N that are incompatible with this model because buoyancy would no longer serve as a restoring force Meeting a surface boundary condition Interaction with the surface of the Earth would influence wave structure and propagation. A minimal boundary condition is that w is zero at the surface. This condition places constraints on other variables through Eqs (1), (3), (4) and (5). For a layer near the surface of the Earth, Eq. (4) specifies that the buoyancy s must be approximately constant in time if w is always near zero. If the only disturbances allowed in the model are cyclic and propagating, it follows that s must be near zero. The time rate of change of w is small even away from the surface and would be even smaller where w is constrained to be always near, so that Eq. (3) reduces to 2 u = 1 ρ p z. (27) Thus near the surface, the non-traditional Coriolis term associated with the zonal wind balances the vertical pressure gradient force term and is much larger than buoyancy. This result suggests that on approach toward the surface, a wave in nature would show progressively smaller vertical displacements and more pronounced contributions from the non-traditional Coriolis terms. Although it is difficult to find a single solution that describes the evolving structure of a wave that propagates between the middle troposphere and the surface of the Earth and meets this boundary condition, we can assess the contributions of this Coriolis term by adjusting model parameters in a manner that amplifies the effect of the term and then compare the solutions with wave structures observed near the surface. We now seek an approximation for the effects of such reduction of w by adjustment of the model parameter N 2. Since the only appearance of N 2 in the governing equations is in a product with w in Eq. (3), adjusting N 2 downward can produce similar effects on other variables as constraining w. Solutions assuming small N 2 must have much smaller vertical velocities than those achieved in solutions with large N 2 values because the total force acting on air parcels in the vertical is much lower. A layer at the surface of the Earth might evolve as if N 2 were effectively zero, because a surface that does not move in the vertical has an infinite buoyancy period. If this reduction in N 2 were the only constraint on w, and if we considered propagation across multiple layers, energetic waves propagating from above where the effective N 2 is larger would yield large amplitude displacements as the restoring force weakens toward the surface. Thus the analogy to decreasing N 2 toward the surface is not perfect. The actual constraint of the boundary condition is directly on w. Nevertheless, analysis of observed signals (below in section 3) suggests that MRG waves proximate to the surface of the Earth are similar to solutions that assume small values of effective N 2. We thus assume that the impact of proximity to the surface of the Earth is analogous to a gradual decrease in the effective N 2 experienced by a large-scale wave toward the surface. Some of the impacts of reduction of the effective N 2 can be assessed from the solutions and compared with observations. According to (2), for a given ν and k, l varies with N 2. Thus if we know the effective N 2 for a given layer, given ν and k, we can find the corresponding value of l.the converse is also true. Equation (2) can be restated ( k 2 kβ ν ) ( = (2n + 1) βl (N 2 ν 2 ) 1/2 ) l2 ν 2 N 2 ν 2. (28) If we hold k and ν constant (thus making the left-hand side constant) and allow N 2 to approach ν 2, the vertical wave number l must also approach zero to maintain equality. If

6 Analysis of Vertically Propagating Convectively Coupled Equatorial Waves 19 l approaches zero, it implies that the wave planes approach vertical. A more careful analysis of a WKB approximation (e.g. Kundu, 199) to Eqs 1 6 might provide greater insight about the details of propagation across different layers, but it is beyond the scope of this paper. 3. Diagnosing model parameters consistent with observed waves We analyse observational data in the context of (7), (8), (17) and (2) to estimate the values of α near particular layers of the lower to middle troposphere associated with a subset of observed convectively coupled waves. We now consider the above Boussinesq solutions to be linearizations of the observed convectively coupled waves about specific layers of the troposphere. The approximate parameters and the model solutions allow us to estimate the zonal phase shifts attributable to the non-traditional Coriolis terms, which can be compared with similar phase shifts in observed waves. We apply wave-number frequencywavelet spectrum analysis (Kikuchi and Wang, 21) along with simple linear regression (Wheeler et al., 2) to diagnose the structures of MRG waves at specific wave numbers and frequencies for direct comparison with the theoretical patterns. The corresponding spatial structures and vertical tilts are measured directly from regressed waves (Wheeler et al., 2) Data Daily interpolated outgoing longwave radiation (OLR) data on a 2.5 grid are applied as proxy for moist deep convection (Liebmann and Smith, 1995). These OLR data have been updated every few months since 1995 following the original algorithm. Daily mean zonal and meridional wind, temperature, specific humidity and geopotential height data are obtained for standard pressure levels from the US National Center for Atmospheric Research/National Centers for Environmental Prediction(NCAR/NCEP) reanalysis (e.g. Kalnay et al., 1996). The mean and first four harmonics of the seasonal cycle are subtracted from the OLR and wind data to generate anomalies. All data are analysed for the period 1 June through to December 29, except that OLR data are missing during 17 March through to 31 December Prior to wavelet analysis, in order to reduce data storage and computation requirements, we first average OLR anomalies between 15 Sand15 N, filtered by symmetry across the Equator. Symmetric OLR signals are obtained by adding OLR signals at the same latitudes across the Equator, and antisymmetric signals were obtained by subtracting the signals, as by Wheeler and Kiladis (1999) Space time wavelet decomposition Wheeler and Kiladis (1999; hereafter WK99) and many others have demonstrated that signals obtained from the regions of the wave number frequency domain near equatorial beta-plane shallow-water model dispersion curves of 25 m equivalent depth are associated with spatial patterns similar in some respects to those obtained from the shallow-water theory. To provide context, Figure 2 shows dispersion lines of 8 and 9 m equivalent depths from the shallow-water model superimposed on an OLR Figure 2. Wavenumber frequency spectrum of OLR anomalies from 15 N to 15 S. Results are normalized by dividing by an estimated red background. Dark shades represent power above the background. Dispersion curves from equatorial beta-plane shallow-water theory are included for reference, for equivalent depths of 8 and 9 m. spectrum (calculated by Fourier transforms in a similar manner to WK99, normalized by dividing by the same spectrum smoothed by 4 applications of a filter). All subsequent observational analyses in this study are reported for MRG waves at the 25 m equivalent depth, nearly between the 8 and 9 m curves in Figure 2, near the maximum power in the MRG band. The above theoretical analysis provides the corresponding space time structures of Boussinesq wave solutions for specified zonal wavenumbers and frequencies. Most previous observational analyses of wave structures, in contrast, conglomerate structures of a broad range of wavenumbers and frequencies associated with the waves (e.g. Wheeler et al., 2). Interpretation of such results is complicated because the vertical and meridional structures of the observed waves might vary with wavenumber and frequency. Thus, clear comparison with the theoretical patterns requires extraction of observed wave signals associated with specific wavenumbers and frequencies. We apply zonal wave number-frequency wavelet analysis to extract such signals from OLR data. A detailed description of space time wavelet analysis is beyond the scope of this paper, but Kikuchi and Wang (21) and Wong (29) offer overviews of the technique. The space time wavelet transform is the wavelet transform in longitude of the wavelet transform in time of the antisymmetric 15 Sto15 N mean OLR anomalies. We apply the Morlet wavelet ψ(s) = 1 (πb) 1/2 exp(iσ s)exp( s2 /B) (29) where s represents x or t for the spatial or temporal transforms, respectively, and σ represents angular frequency ν or wavenumber k., The bandwidth parameter B was assigned a value of 2/k for the spatial transform and 5/ν for the temporal transform. Conclusions are not sensitive to these arbitrarily assigned values of B, but much larger values reduce the amplitude contrast of signals and enhance a ringing effect, and substantially smaller values do not sufficiently resolve large-scale or low frequency waves. The transform is obtained by taking the time-centered dot product of the wavelet and all daily consecutive overlapping time series segments at each grid point around the globe, then applying a similar transform to the result in longitude

7 11 P. E. Roundy and M. A. Janiga by the same approach. The final result is calculated at every day and longitude grid point in the OLR data from 1975 to 29, inclusive. The product of the result and its complex conjugate, averaged in time and longitude, yields a power spectrum similar to that of WK99, except near wavenumber zero, where amplitude is lost (not shown, but see Wong, 29). We limit our analysis to wave numbers 3 and higher to avoid the apparently degraded signals at the largest spatial scales. Wavelet coefficients were calculated for zonal wavenumbers 3 2 and for periods consistent with the original frequency resolution of WK99, which include all harmonics of 96 days. For convenience, we select frequencies at a given wave number along the dispersion curves near the peaks in the normalized spectra of WK99, including 12, 25 and 9 m equivalent depths, but for brevity, we report only results for the MRG wave at h = 25 m Linear regression models Simple linear regression is applied to diagnose coherent structures that tend to be associated with observed wave signals (e.g. Hendon and Salby, 1994; Wheeler et al., 2). Using the space time wavelet transform at a selected wavenumber and frequency, we extract a base index time series for each grid point over a range of longitudes. Either the real or imaginary parts of the transform work for the base index. We chose the imaginary part for convenience of the associated zonal phasing, but the conclusions are the same regardless of this choice. We select the geographical points from the region that shows the greatest OLR variance in the wavenumber frequency band of the MRG wave (Wheeler et al., 2; Roundy and Frank, 24), from 16 Eto17 W. The time series from each of those points serve as predictors in regression models at each grid point across a broad geographical domain to diagnose the associated structures. One grid of regression models is calculated for each base point time series. To illustrate, we model the variable Y at the grid point S as Y s = P x A s, (3) where P x is a matrix whose first column is a list of ones and second column is the base index at the longitude grid point x. A s is a vector of regression coefficients at the grid point S. After solving for A s at each grid point by matrix inversion, (3) is then applied as a scalar equation to diagnose wave structure by substituting a single value for the second column of P x that is representative of a crest of a wave located at the base longitude (we set its value at +1 standard deviation). These regression models are applied to create composite anomalies of OLR, u and v winds, and density across the global tropics at standard pressure levels. Results are calculated for the region 9 to the east and west of each base longitude, and then the set of results from all base points are averaged to reduce noise and geographic effects. Corresponding regressed geopotential heights are also calculated, and results are used in plotting the other regressed data to simplify comparison with the Boussinesq solutions, which are expressed in terms of height instead of pressure. The statistical significance of the difference from zero of the result at each point on the map is assessed based on the correlation coefficient (e.g. Wilks, 25), and we analyse and discuss only those regressed signals that are deemed to be significantly different from zero at the 9% level. Since the regression is accomplished in the time domain, some signal from wavenumbers other than the target wavenumber can appear in results. However, we found such contributions to be small in comparison with signals at the target wavenumber Comparison of observations with the Boussinesq wave solutions We estimated the best fit of the Boussinesq wave solutions to observed waves by measuring the characteristics of regressed waves and substituting the resulting parameters into the model. We then applied the model to predict other characteristics of the regressed waves, and we compare those characteristics with direct measurement of the regressed waves. The zonal wave number k and the frequency ν are specified in the regressed waves. l is estimated by measuring the vertical slopes in the longitudinal direction of anomalies of the lower to mid-troposphere in the equatorial vertical cross-sections of meridional wind in a regressed MRG wave, then by multiplying the result by the corresponding value of k. Meridional wind was chosen because of its strong signal at the Equator relative to other variables. The reanalysis data do not allow us to directly measure the vertical slopes of the waves close to the surface. We applied time-lag regression analysis of sounding data from Tarawa island (1.35 N, E) to examine whether MRG wave planes approach vertical near the surface of the Earth. N 2 is calculated from temperature and specific humidity data from the NCEP/NCAR reanalysis, and is assumed to be everywhere equal to its value averaged over the troposphere from 85 to 2 hpa and from 1 Nto1 S (.323 s 1 ). n is set equal to for MRG waves. Given these values of k, l, ν, n and N 2, we find the solution of Eq. (2) for α (given 26) that yields a wave structure consistent with the selected value of n. After tuning the model to these observed parameters, we check whether the upward motion implied in the solutions is consistent with the locations of the most negative OLR anomalies in the corresponding regressed waves. We further check that model vertical velocities are consistent with regressed vertical velocities in the NCEP/NCAR reanalysis. Although we have low confidence in the specific magnitudes of the reanalysis vertical winds, agreement between those winds and our results might increase confidence. Lastly, we estimate the magnitude of the phase shift δ(y) at±2 latitude associated with the inclusion of the non-traditional Coriolis terms from the model, given measured values of l. To verify these predictions, we measured zonal phase shifts from the regression maps by first finding the longitude of the equatorial centre of a gyre. Then, at 2 N within the same gyre, the corresponding phase of the wave is the point of minimum absolute value in the meridional wind. The phase shift is calculated from the zonal distance between the point on the Equator and the point at 2 N. 4. Results: observed and Boussinesq MRG waves The horizontal structures of regressed MRG waves including OLR anomalies (shaded) and 1 hpa winds are plotted in Figure 3(a) (d) for k = 3 through to k = 6, respectively. Positive and negative OLR anomalies form patterns that are antisymmetric about the Equator. Convective anomalies

8 Analysis of Vertically Propagating Convectively Coupled Equatorial Waves 111 associated with MRG waves originate within the eastern sides of the cyclonic portions of the circulations and extend poleward and eastward across part of the low-level anticyclonic portions of the wave (consistent with Liebmann and Hendon (199) and Kiladis et al.(29)). Circulation patterns are also highly distorted toward the east with latitude. Heavy black lines are drawn on Figure 3 to highlight this distortion. The north south lines locate the equatorial centres of selected gyres. Heavy slanted lines approximate the trough or ridge axes, defined as the longitude within the gyre at a given latitude at which the meridional wind is closest to zero. These results suggest that at 1 hpa, zonal phase shifts of roughly 25% of the wavelength are observed. Figure 4 shows regressed horizontal and NCEP/NCAR reanalysis vertical winds (shading) at 85 hpa. Comparison of the horizontal wind patterns with the corresponding patterns in Figure 3 suggests that meridional tilts in the horizontal wind structures are much less pronounced near 85 hpa than at 1 hpa. Some more substantial meridional tilts are apparent on wavenumber 5 and 6 waves or on the western and southern portions of the domains. Measureable eastward shifts in the horizontal winds near the centre of the domain are at most a few per cent of the wavelength in the Northern Hemisphere, larger in the Southern. We also analysed wave structure near 5 hpa (not shown), and results suggest even less distortion of the horizontal winds, but the vertical wind anomalies show more meridional tilt in comparison with the horizontal wind anomalies. Some of the distortion in the vertical wind anomalies observed at 85 hpa and above might be forced by convection originating from the more distorted horizontal wind patterns observed closer to the surface of the Earth. Figure 5 shows the vertical structure of the regressed MRG wave at zonal wavenumber 4 along 7.5 N, including v and ρ in panel (a) and v and w in panel b. Regressed MRG waves at other wave numbers are similar, but with wave planes characterized by different slopes in the vertical (not shown). Negative (positive) ρ anomalies occur with southward (northward) flow. Anomalous upward motion occurs roughly in quadrature with anomalous ρ and v, with maximum upward motion immediately to the west of the region of anomalously dense air. Figure 6(a) (d) shows the composite longitude height cross-sections for equatorial meridional wind, with northward flow shaded, for wave numbers 3 6, respectively, for comparison with Figures 3 and 4. Anomalies in the eastern portion of the domain tilt eastward with height through the troposphere, become more vertical near the tropopause, and tilt westward with height through the stratosphere. Anomalies across the western portion of the domain show distinctly opposite signs between the lower and upper troposphere instead of tilted structures. These vertical dipole patterns are consistent with the first baroclinic vertical mode, or with the structure of merdional flow associated with tropical cyclones, from which MRG waves might disperse eastward. Such patterns might also suggest that the atmosphere becomes unstable to moist deep convection on the spatial scales of those waves, implying failure of (2). Lines drawn through the tropospheric anomalies in Figure 6 are used to estimate vertical tilt, which is used to find l as discussed in section 3.4. These lines are not intended to provide specific values, but instead outline a reasonable range of values representative of the observed waves near particular altitudes. Changes in slope near the surface are not well resolved in Figure 6, but a time-lag regression analysis of Tarawa meridional wind sounding data suggests that the slope does approach the vertical proximate to the surface (not shown). When the slope is nearly vertical, small changes in the slope yield large changes in the estimated value of l. Thus direct measurement can yield no more specific conclusion than that the vertical wavelength exceeds 1 5 m. For such slopes, we noted first that the slope was nearly vertical (consistent with the theory for wave planes near the surface), then we estimated the more specific value of l from Eqs (17) and (18) by first measuring the zonal phase shifts in the regressed horizontal wind anomalies at 1 hpa. Table I shows parameter values estimated from regressed waves and the corresponding predictions of α and δ based on the theory (with the exception of values at 1 hpa, for which vertical wavelength is made more precise by applying the theoretical result after measuring δ). Values from Table I are substituted into the Boussinesq solutions for the MRG wave to obtain the results plotted in Figures 7 and 8 for 1 and 85 hpa, respectively. Only the wavenumber 4 MRG wave is plotted because the results for other wavenumbers are similar. Shading in Figures 7 and 8 represents upward motion. These solutions compare favourably with the similar OLR and w anomalies in Figures 3 and 4(b). The Boussinesq solutions show similar phase relationships between u, v, w and ρ anomalies to those shown in Figure 5. Some of the similarity between the tropospheric portions of the cross sections of the meridional wind anomalies in regressed MRG waves east of the dateline and the Boussinesq solutions for meridional wind is by design, since the parameter l was measured from the regressed waves and k is predefined. However, these parameters do not individually predetermine the meridional width of the solution in Figures 6 and 7, nor do they individually predetermine the zonal phase relationships between solutions for u, v and w. Correspondence between vertical wind anomalies in the solutions and regressed OLR anomalies of opposite sign suggest that the Boussinesq waves are good analogues to the regressed observed waves, especially near the centre of the composite domain where results are most reliable. The vertically tilted solutions of the Boussinesq waves show similar phase relationships at a given height between the winds and buoyancy to the patterns shown from observations and reanalysis data in Figure 5. Accounting for vertical variation in α, the theory is consistent with the observation that upward motion begins at lower levels in cyclonic flow anomalies in MRG waves, and this upward motion tilts eastward, reaching a maximum at mid-levels over the regions of strongest poleward low-level flow. Where wave planes are nearly vertical proximate to the surface, the theory predicts that vertical motion vanishes, consistent with the surface boundary condition. Ascent in the upper levels occurs over low-level anticyclonic flow because of vertical tilt through the mid-levels. These results suggest that the vertical structures of observed waves are similar to the Boussinesq MRG wave, taking into account the varying influence of convection parameterized in α and proximity to the surface (here also parameterized in α). The most negative OLR anomalies occur at roughly the same point in the horizontal wave structure as low to mid-level upward motion in the Boussinesq wave. The tilted vertical motion in the Boussinesq solution is consistent with a progression from shallow convection beginning within the low-level cyclonic

9 112 P. E. Roundy and M. A. Janiga (a) MRG 1 hpa Winds and OLR, h=25 k=3 (b) MRG 1 hpa Winds and OLR, h=25 k=4 1S 1S 2S S (c) MRG 1 hpa Winds and OLR, h=25 k=5 (d) MRG 1 hpa Winds and OLR, h=25 k=6 1S 1S 2S Longitude: Degrees East of Base Point 2S Longitude: Degrees East of Base Point Figure 3. Regressed OLR (shading) and vectors for the horizontal wind (1 hpa), based on wavelet analysis of OLR anomalies at specific wavenumbers within the MRG band along the shallow-water dispersion curve at the equivalent depth of 25 m. Negative OLR anomalies consistent with enhanced moist deep convection are shaded. The contour interval is 1 W m 2. Panels (a) -(d) show results for zonal wavenumbers 3 6, respectively. The longest wind vector is roughly.5 m s 1. Anomalies are small relative to regressed MRG waves reported by Kiladis et al.(2) because their results included signal from a broad range of neighbouring wave numbers and frequencies. (a) MRG 85 hpa Winds, h=25 k=3 (b) MRG 85 hpa Winds, h=25 k=4 1S 1S 2S S (c) MRG 85 hpa Winds, h=25 k=5 (d) MRG 85 hpa Winds, h=25 k=6 1S 1S 2S Longitude: Degrees East of Base Point 2S Longitude: Degrees East of Base Point Figure 4. Regressed NCEP/NCAR reanalysis of w wind (shading) and vectors for the horizontal wind, based on wavelet analysis of OLR anomalies at specific wavenumbers within the MRG band along the shallow-water equivalent depth of 25 m. For comparison with Figure 3, wind anomalies are plotted at 85 hpa instead of 1 hpa. Upward motion is shaded. The contour interval is.2 cm s 1, with contours beginning at.6 cm s 1.

10 Analysis of Vertically Propagating Convectively Coupled Equatorial Waves 113 Figure 5. Vertical structure of the regressed MRG zonal wavenumber 4 MRG wave at 7.5 N. Panel (a) shows v wind (shaded) with density anomalies contoured (positive in red, with the contour interval on the top right). Panel (b) shows v wind (shaded), with NCEP/NCAR reanalysis vertical wind in contours (anomalous upward motion in red, with the contour interval listed on the top right). k is expressed in terms of global wavenumber. Figure 6. Regressed equatorial v wind anomalies plotted with respect to height corresponding to the MRG-wave regression results shown in Figures 3 and 4. Lines drawn on some anomalies were used to estimate vertical slope, and resulted in the estimates for vertical wavelength written on the diagrams (in m). Positive values are shaded, the zero contour is omitted, and the maximum amplitude is close to.5 m s 1. The horizontal axis represents degrees of longitude with positive values east of the base longitude. flow, to deep convection located in the vicinity of low-level poleward flow, to stratiform and cirrus in the vicinity of lowlevel anticyclonic flow. These patterns are consistent with previous observations of the convectively coupled MRG wave (e.g. Kiladis et al., 29). 5. Analysis of the non-traditional Coriolis terms A first glance at Eq. (2) reveals that this dispersion relationship does not depend on the non-traditional Coriolis terms (consistent with Fruman, 29). Although (2) does not

11 114 P. E. Roundy and M. A. Janiga Table I. Parameter values for MRG waves characterized by specific zonal wavenumbers and vertical wavelengths. The periods and zonal wavenumbers are consistent with a shallow-water model equivalent depth of 25 m. The first two vertical wavelengths listed from top to bottom for each zonal wavenumber correspond with the slopes measured from the lines plotted on the regressed waves in Figure 4. The third vertical wavelength is estimated from Eqs (17 and 18) assuming the zonal phase shifts evident in Figure 3 for 1 hpa flow. The fifth column shows the fractional change in the equivalent depth parameter between the solution to the hydrostatic equations and the solution to the non-hydrostatic equations Zonal Period Vertical α δ(2 N, as wavenumber (days) wavelength ( 14m) fraction of wavelength) ( hnon hydrostatic ) h hydrostatic h non hydrostatic (85 hpa) (5 hpa) (1 hpa) (85 hpa) (5 hpa) (1 hpa) (85 hpa) (5 hpa) (1 hpa) (85 hpa) (5 hpa) (1 hpa) Latitude u, v, and w at 96 m (1 hpa), n=, k=3.8, vertical wavelength = 5e+5 m Longitude Figure 7. u and v winds (vectors) and w at 96 m for MRG wave solutions near zonal wavenumber 4 and a vertical wavelength of m. Latitude u, v, and w at 156 m (85 hpa), n=, k=3.7, vertical wavelength = 4.5e+4 m Longitude Figure 8. Same as Figure 7, except for at 85 hpa, assuming the vertical wavelength m. include these terms, the structural solutions in Eq. (23) do. A tracer of dependence on these terms is the explicit appearance of (i.e., ignoring appearances of in β). appears explicitly in both Ɣ and K. Its appearance in K demonstrates that these terms modify the meridional widths of the wave structures. The explicit dependence in Ɣ results in a zonal phase shift in the wave structures that depends on latitude, l, N 2 and ν that is expressed as an eastward tilt with distance from the Equator (consistent with Fruman, 29). For example, the poleward ends of circulations in MRG waves occur to the east of the zonal centre of the disturbances on the Equator. Dependence of Ɣ on ν is a result of relaxation of the hydrostatic approximation and thus was excluded from the result of Fruman (29). Ɣ varies more than a few per cent with ν in moist waves near the surface of the Earth only for periods shorter than about 4 days, but only if the effective N is smaller than about 1 4 s 1. Thus the hydrostatic approximation is applicable to the full range of waves analysed here. Explicit dependence of K on disappears when (1 α)n 2 ν 2 >> 4 2. By this interpretation, the nontraditional Coriolis terms might be irrelevant to dry waves unless the wave frequency is close to the buoyancy frequency, except proximate to the surface of the Earth. Further, reduced moist stability or close proximity to the surface are required to generate values of the effective N 2 sufficiently small for these Coriolis terms to be relevant to lower frequency convectively coupled waves. As N 2 approaches Ɣ, K approaches zero, implying that the equatorial trapping radius goes to zero, and the solutions to Eq. (2) become undefined. Nevertheless, the solutions indicate a broad range of parameter values over which the impacts of the non-traditional Coriolis terms matter, and the analysis of observations demonstrates the anticipated impacts. Observed vertical tilts of regressed waves characterized by particular wavenumbers and frequencies suggest values of parameters representative of observed waves that evolve in a manner consistent with our original assumption of static stability on the spatial scales of the waves. These parameters are applied in Table I to diagnose the magnitudes of Ɣ and the zonal phase shift δ associated with observed waves. Values of Ɣ are of order 1 13 m 2. Zonal phase shifts associated with such values of gamma would be zero on the Equator and would range from roughly.1