a. Sounding data

The sounding data used for this study were collected during the intensive operating period (IOP) of TOGA COARE, which was from 1 November 1992 through 28 February 1993. For a detailed description of this experiment and a map of the sounding stations see Webster and Lukas (1992). Radiosonde, wind profiler, and surface data were objectively analyzed onto a grid (Ciesielski et al. 1997, 2003) with a vertical resolution of 25 hPa, a temporal resolution of 6 h, and a horizontal resolution of 1°. Many of the plots we present are average vertical profiles for grid points within the Intensive Flux Array (IFA), which was an array of sounding stations centered near 2°S, 155°E.

b. Satellite data

Gridded measurements of infrared brightness temperature (T_B) by the Geostationary Meteorological Satellite were obtained at a resolution of ¹/₁₀° × ¹/₁₀° over 15°S–15°N, 130°E–180° and averaged in 1° × 1° bins.

c. Compositing technique

To obtain the composite 2-day wave we simply temporally filter (Duchon 1979) the average T_B over the IFA, retaining fluctuations between 1.5 and 3 days (Haertel and Johnson 1998, Fig. 5), and then project dependent variables (such as temperature, wind, and moisture) onto the filtered T_B at a series of lags (e.g., Wheeler et al. 2000; Straub and Kiladis 2002). In this linear approach the amplitude of the independent variable is assigned an arbitrary value. In the examples used here the wave is scaled to a −20°C perturbation in IFA T_B at zero lag, a typical value during the peak phase of a strong 2-day wave (Takayabu et al. 1996, Fig. 4; Haertel and Johnson 1998, Fig. 6).

d. Vertical mode decomposition

For the two mode simulation (section 5) we project the 2-day wave's heating onto the first two linear vertical modes (Fulton and Schubert 1985) of an atmosphere with a rigid upper boundary at 150 hPa. The vertical structure functions for the modes are calculated for the average IFA sounding using the method described by Haertel and Johnson (1998).

a. Local cloudiness and rain rate

The composite IFA T_B has local minima at −48, 0, and 48 h (Fig. 2a). While one would expect such a pattern given the compositing technique, it might also reflect the tendency of 2-day waves to occur in series, which has been well documented in previous studies (e.g., Takayabu 1994b; Takayabu et al. 1996; Clayson et al. 2002). The T_B signal appears to be a good proxy for moist convection; the budget-derived perturbation rain rate is roughly out of phase with T_B, and has a similar relative amplitude at all lags (Fig. 2b). The minimum T_B lags the maximum rain rate slightly, which likely reflects the time it takes a cirrus anvil to develop from convective towers. The amplitude of the rain-rate perturbation is comparable to the mean rain rate for the IFA for the IOP (Johnson and Ciesielski 2000).

b. Horizontal structure and propagation

The composite wave has a scale of about 1000–1500 km (Figs. 3a–e). It originates more than 1000 km to east of the IFA when a large-scale disturbance divides into northward- and westward-propagating components. The latter intensifies as it passes over the IFA and dissipates about a day later. The wave propagates at about 16 m s⁻¹. Upper-level wind anomalies diverge from the negative T_B anomalies and converge into positive T_B anomalies, supporting the interpretation of T_B as a proxy for deep convection.

Throughout the wave's life cycle the high cloudiness exhibits a weak signal of zonal periodicity with a wavelength of about 25° longitude (e.g., Figs. 3a,c). The wavelike structure propagates westward within a stationary amplitude envelope centered on the IFA (the base point for the composite). The horizontal structure of the composite is similar to that of the composite 2-day wave constructed by Takayabu (1994b), which is an average of 263 cases that occurred between 1980 and 1989 in the boreal winter in the same region. However, Takayabu's composite lacks the northward-propagating disturbance in the vicinity of 170°E, so it is possible that this portion of the composite represents a peculiarity of the 1992–93 season.

c. Vertical structure

The composite 2-day wave is accompanied by changes in temperature, moisture, heating, moistening, winds, and divergence through most of the troposphere over the IFA. These changes are detailed below.

a. The numerical model

The numerical model solves the inviscid primitive equations on a β plane linearized about a basic state of rest:

View Expanded

where x and y represent horizontal displacement, p is pressure, t is time, u is perturbation zonal velocity, υ is perturbation meridional velocity, β = ∂f/∂y (where f is the Coriolis parameter), T is perturbation temperature, R is the gas constant for dry air, ϕ is perturbation geopotential, ω is perturbation vertical velocity, S = −T(∂ logθ/∂p) (where θ is potential temperature and the overbars denote basic-state variables that are functions of p only), and Q is perturbation heating. This system of equations was selected because its solutions are algebraically equivalent to those of classical equatorial wave theory (Matsuno 1966; Lindzen 1967; Fulton and Schubert 1985), which previously has been used to interpret the observed horizontal structures and dispersion characteristics of convectively coupled tropical waves (Takayabu 1994a,b; Wheeler and Kiladis 1999; Wheeler et al. 2000).

Equations (1)–(5) are approximated using fourth-order spatial differencing with Δx = Δy = 100 km, second-order vertical differencing with Δp = 6.25 hPa, and leapfrog time differencing with Δt = 180 s. Vertical velocity is set to zero at both the lower (1000 hPa) and upper boundaries (6.25 hPa). The high location of the upper boundary allows vertical propagation of gravity waves with minimal reflection into the domain of interest. Horizontal boundaries are also placed sufficiently far from domain of interest (at x = ±20 000 km and y = ±5000 km) that they have a negligible impact on the solution. The external mode is filtered from the solution to allow the use of a long time step.

Analysis of average absolute vorticity fields for the COARE IOP suggest that through most of the troposphere the dynamical equator lies between 1° and 2°S (similar to what Takayabu 1994b observed), which is very close to the latitude of the center of the IFA (2°S). Hence, simulated vertical profiles over the model origin (x = 0, y = 0) are compared to observed profiles over the IFA.

b. Constructing the heating function

We define the heating to have the following form:

Qx,y,p,tQ_υpQ_ttQ_hx,yQ_wx,t(6)

where Q_υ, Q_t, Q_h, and Q_w denote the vertical structure, temporal envelope, horizontal envelope, and propagating wave component of the heating, respectively. The vertical structure function Q_υ (Figs. 5a–b) is a complex valued function having an amplitude and a phase that come directly from the composite heating analysis (Fig. 4c). The amplitude simply equals the maximum observed heating amplitude at each pressure level. The phase is obtained by testing phases of 1°, 2°, 3°, … , 360° for each pressure level and selecting the one that produces the smallest deviation in heating from the observations (in order to obtain a continuous phase curve in Fig. 5b phases greater than 120° are plotted as negative equivalents). The temporal envelope is defined to be half of a sine wave:

View Expanded

where τ = 6 days (the simulation is run out to t = τ). This function is intended to mimic the temporal dependence of the composite wave's heating (Fig. 4c). While the decay of the latter at large lags results from a temporal decorrelation of the 2-day wave signal and not from the temporal envelope of heating amplitude; such a decorrelation and decay also exist for the wind and temperature perturbations that we are attempting to reproduce.

The horizontal envelope is Gaussian in both x and y:

View Expanded

where r_x = 1400 km and r_y = 500 km. These scales are consistent with those of the composite wave as shown in the T_B analysis (Fig. 3). The propagating wave structure has the form

View Expanded

where λ_x = 2800 km and c = 16 m s⁻¹ are also consistent with the brightness temperature analysis.

c. Comparing the simulation to observations

The time–pressure series of heating over the origin of the model domain (Fig. 6a) closely resembles that observed above the IFA (Fig. 4c), indicating that (6)– (9) adequately represent the observed heating. The simulated temperature profile (Fig. 6b) shows most of the structure that appears in the composite temperature analysis (Fig. 4a): lower-tropospheric temperature perturbations are out of phase with upper-tropospheric temperature perturbations, amplitudes are a few tenths of a degree Celsius, and oscillations are weak or absent around 500 and 900 hPa. There are local minima/maxima near the surface and just below the melting level (∼575 hPa). Tilted structures indicate vertical wave propagation near the tropopause. The degree of similarity between the observed and simulated temperature perturbations is remarkable and is strong evidence in support of our null hypothesis.

The simulated time–pressure series of zonal wind (Fig. 6c) also exhibits most of the structure that appears in the composite zonal-wind analysis (Fig. 4e); each field contains wavelike perturbations that rise with time through most of the troposphere and descend with time near the tropopause. For both cases the oscillations have local amplitude maxima near the surface, at midlevels, and near 200 hPa. Simulated and observed amplitudes are similar at mid and upper levels, but near the surface the simulated wind field is 2–3 times more intense than the observed field. This difference is likely a consequence of the fact that the simulation is inviscid whereas boundary layer turbulence damps near-surface winds in nature. At most pressure levels the phasing of the wind perturbations in the simulation is similar to that in the observations except for just above and just below 200 hPa, and near 700 hPa. Since Q is symmetric about y = 0, along y = 0 (and over the model origin) there is no meridional wind perturbation in the simulation.

The horizontal structure of the simulated wind field is also similar to that of the observed field. At 0 h the simulated winds diverge from the center of the wave and turn toward the right (left) to the northeast and southwest (northwest and southeast) of the wave (Fig. 7a). The divergence of winds and turning is similar in the observed wind field (Fig. 3c). At 12 h the simulated winds diverge from the decaying wave (centered near x = −700 km, y = 0) and arc around to converge into a suppressed region to the east of the wave (Fig. 7b). Once again the simulated wind structure is similar to that observed (Fig. 3d). For both times the amplitudes of the simulated winds are comparable to those of the observed winds (Figs. 3c,d; 7a–b).

The simulation reproduces most of the qualitative structure of the wind and temperature perturbations present in the composite analysis, and the amplitudes of simulated perturbations are generally within a factor of 2 of the amplitudes of the observed perturbations. This result supports the null hypothesis that these perturbations can be modeled as linear responses to convective heating and cooling, and it also suggests that the dynamical structure of the simulated wave is realistic.

a. Vertical mode decomposition

A solution to the primitive equations linearized about a basic state of rest with a rigid upper boundary may be decomposed into a superposition of vertical modes, each of which is algebraically equivalent to a solution of the linearized shallow-water equations (e.g., Fulton and Schubert 1985). For a given vertical mode, profiles of geopotential and winds are proportional to a vertical structure function that is an eigenfunction of a differential operator. Typical vertical structure functions resemble deformed cosine functions, and their exact shapes depend on the temperature profile of the basic state. Each vertical mode is associated with an equivalent depth, or gravity wave speed, that identifies the mode with the shallow-water system (having that depth or gravity wave speed) to which the mode is algebraically equivalent.

In an unbounded atmosphere there is a continuum of vertical modes (and equivalent depths). The practice of approximating the dynamics of such an atmosphere using vertical modes for a bounded atmosphere is analogous to approximating a continuous Fourier transform with a discrete transform. Each mode of the resulting discrete system represents a spectral band of vertical modes of the continuous system, and solutions of the discrete system may be regarded as approximations of solutions of the continuous system (e.g., Mapes 1998). The accuracy of the approximation depends on the positioning of the upper boundary, with a higher upper boundary leading to a greater accuracy. However, in some cases, reasonable solutions may be obtained by placing the upper boundary at the lowest possible position, which is immediately above the prescribed forcing. For example, Pandya et al. (1993) show that for such a positioning of the upper boundary above a forcing characteristic of a mesoscale convective system, the buoyancy and wind response of the bounded atmosphere are quite similar to those of the unbounded atmosphere. One advantage of using such a maximal discretization (or minimal decomposition) is that it simplifies the solution and its interpretation (e.g., Nicholls et al. 1991; Haertel and Johnson 2000).

The simulated 2-day wave's perturbation heating is significantly nonzero only below 150 hPa (Fig. 4c). Therefore, we can obtain a minimal decomposition of the heating by projecting it onto the vertical modes for an atmosphere having the COARE mean temperature profile and an upper boundary at 150 hPa (Fulton and Schubert 1985). The heating projects strongly onto just two modes. One, which is often referred to as the first baroclinic mode, has a gravity wave speed of 49 m s⁻¹ and is associated with deep convective heating of the same sign throughout the troposphere peaking at mid levels (Fig. 8a). The second baroclinic mode (Fig. 8b) has a gravity wave speed of 23 m s⁻¹ and is interpreted as stratiform heating of opposite sign in the lower and upper troposphere (Mapes and Houze 1995). Note that the heating associated with stratiform precipitation is much weaker than the first mode and in the upper troposphere lags deep convective heating by 6 h. When the two-mode heating is subtracted from the observed perturbation heating (Fig. 4c), the residual is very small in amplitude except for a shallow region slightly exceeding 1°C day⁻¹ near the surface (not shown).

b. The response of a bounded atmosphere to the two-mode heating

We now present a simulation in which the heating function described in section 4b is projected onto the two vertical modes described above and applied to an atmosphere having a rigid upper boundary at 150 hPa. The two-mode simulation reproduces most of the structure of tropospheric wind and temperature perturbations present in the composite analysis. This result supports the use of two vertical mode models for analyzing the dynamics of convectively coupled equatorial waves.

a. Testing the hypothesis

One of the objectives of this study is to test the null hypothesis that the wind and temperature perturbations associated with 2-day waves can be modeled as linear responses to convective heating perturbations. To the extent that Figs. 4 and 6 display reasonable agreement, given both observational error and limitations of the technique used to isolate the observed signals, we conclude that this hypothesis cannot be rejected based on our test.

While there are limitations to this test that might provide a bias toward not rejecting the hypothesis (e.g., the heating and temperature/wind fields are not independent datasets), there are several ways the simulation could have yielded significantly different temperature and wind perturbations that apparently did not occur. First, budget-derived profiles of heating are easily contaminated by convective and mesoscale noise (Mapes et al. 2003). An erroneous heating profile would likely have produced wind and temperature perturbations that did not agree with observations. Second, if 2-day waves were forced by preexisting dry dynamics (e.g., Straub and Kiladis 2003a), only a portion of their wind and temperature would be reproduceable as a response to heating function. Third, if nonlinear dynamics or advection by basic-state winds were fundamental to the generation of the wave's kinematic and thermodynamic structure, the dynamical system that we chose likely would not have reproduced the observations. Hence, while the results presented in section 4 are not a foolproof test of the hypothesis, they do strongly support the hypothesis and instill confidence that the observed heating profile is accurate.

b. The two vertical mode model of 2-day waves

The simulation presented in section 5 shows that the two-mode model captures the basic dynamics of 2-day waves. It represents their tilted heating, wind, and temperature perturbations as the superposition of two simple waves (Fig. 10). This model also sheds light on the origin of the temperature structure (Fig. 4a): deep tropospheric heating and cooling are nearly balanced by adiabatic temperature changes due to vertical motion (Figs. 10b, 11a), but the second vertical mode is more out of balance, resulting in the out of phase lower- and upper-tropospheric anomalies (Figs. 10c, 11b).

Although the vertical modes are dry-dynamical constructs, they appear to represent the moist physical processes of deep convection, shallow convection, and stratiform precipitation. Mapes (2000) associated the first vertical mode's positive heating with deep convection and the second vertical mode's upper-tropospheric heating and lower-tropospheric cooling with stratiform precipitation. However, we emphasize that it is also appropriate to associate the second mode's heating of the opposite sign, upper-tropospheric cooling and lower-tropospheric warming, with shallow convection. The 2-day wave schematic of Takayabu et al. (1996) (Fig. 1) shows shallow convection on the leading edge of the wave that is accompanied by upper-tropospheric subsidence (and presumably radiational cooling). The cloud structure in this schematic is based on, among other things, a histogram of cloud-top brightness temperatures. On the leading edge of the simulated 2-day wave the second vertical mode contributes lower-tropospheric warming and upper-tropospheric cooling that are accompanied by rising motion and subsidence respectively (Fig. 10c), and which match up with the region of shallow convection in Fig. 1. Hence, the two vertical mode model of 2-day waves not only captures the wave's basic dynamics, but it also represents the wave's convective structure as well.

c. An n = 1 westward-propagating inertio–gravity wave?

As we mention in section 1, one of the theories of 2-day waves is that they are n = 1 westward propagating inertio–gravity (WIG) waves. The results presented in this paper are generally consistent with this idea. First, the simulations demonstrate that a dynamical system linearized about a basic state, such as that from which equatorial wave theory is derived, can capture the dynamics of 2-day waves. Second, the horizontal structure of the composite 2-day wave resembles the theoretical structure for an n = 1 inertio–gravity wave. The composite wave has a zonal wavelength of about 2800 km and a phase speed of about 16 m s⁻¹ (Fig. 3). The forced n = 1 WIG wave (Lindzen 1967) with these characteristics has an equivalent depth of 14.3 m. Its horizontal structure is depicted in Fig. 12 (adapted from Matsuno 1966). Velocity vectors diverge from centers of downward vertical motion and arc around and converge into centers of upward vertical motion. Its geopotential height (not shown) and vertical velocity perturbations are symmetric about the equator. The composite 2-day wave has a similar horizontal structure (e.g., Fig. 3d). Of course, the theoretical wave and composite wave differ in that the former has an infinite zonal extent, and the latter is a “wave packet” a few thousand kilometers across, but it appears that their dynamical essence is the same. Finally, the dynamical structure of the 2-day wave is very similar to that of the composite convectively coupled wave constructed by Wheeler et al. (2000, hereafter WKW) that was obtained by filtering for spectral components that lie along the dispersion curve for the n = 1 WIG wave. The patterns of tropospheric temperature, zonal wind, and vertical motion perturbations are nearly identical in the two waves (compare Fig. 10a with Fig. 23 of WKW). The phase speed of the 2-day wave is much slower than the 30 m s⁻¹ phase speed found in WKW, but there are several factors that may be contributing to this discrepancy: 1) The composite WIG wave in WKW has a larger zonal wavelength, so based on the dispersion diagram for the n = 1 WIG waves one would expect it to propagate more rapidly; 2) during COARE the mean flow at low levels was westerly, which would have slowed the 2-day waves; 3) the COARE waves may have been phase locked to the diurnal cycle over the warm pool (Takayabu et al. 1996); and 4) even for a fixed horizontal wavenumber the n = 1 WIG waves analyzed by WKW covered a range of frequencies (see their Fig. 2b), that is, had a range of phase speeds, and one would expect that a composite constructed from a handful of cases might have some differing properties from a composite constructed from more than a decade's worth of data.

One question that remains open for 2-day waves and other convectively coupled equatorial waves is the cause of their shallow equivalent depth. As we mention above, if the composite 2-day wave is an n = 1 WIG wave, it has an equivalent depth of about 14 m, which is significantly different from the “dry” equivalent depths of the first two vertical modes, which are 245 m and 54 m, respectively. Analysis of the balance between Q and Sω in the simulation (Fig. 11) suggests that the former partially cancels the latter, effectively resulting in a reduced static stability. For each mode the degree of cancelation is roughly consistent with the reduction of equivalent depth. For the first vertical mode Q cancels all but about 1/20 of Sω (Fig. 11a), and the equivalent depth is reduced by a factor of almost 20. For the second vertical mode the degree of cancelation varies, but on average Q cancels all but 1/4 of Sω (Fig. 11b), and the equivalent depth is reduced by a factor of about 4. Since the simulated temperature perturbations closely resemble the observed temperature perturbations, we expect that a similar degree of cancelation exists for 2-day waves in nature. But a key question is: what determines the degree to which Q cancels Sω? In the simulation the degree of cancelation between Q and Sω depends on the extent to which the forcing resonates with the waves it excites, which depends on the prescribed phase speed of the forcing (e.g., Haertel and Johnson 2000). However, in nature the opposite causality might exist; that is, physical processes could determine the degree of cancelation between Q and Sω, which in turn could determine the speed of the convectively coupled wave. Theories have been proposed that quantify the reduction of static stability by deep convection (e.g., Neelin and Held 1987; Emanuel et al. 1994) and it has been simulated in models (e.g., Sobel and Bretherton 2003), but we are unaware of any such theories for shallow convection and stratiform precipitation. Of course, it is possible that one of the two vertical modes is driving the propagation of the system and that the other amounts to a forced response. It is also possible that the reduction of equivalent depth results from the the interaction of the two vertical modes (Mapes 2000; Majda and Shefter 2001; Majda et al. 2004). In any event, the results presented here support approaching this problem from the context of a two vertical mode system, which seems to do an excellent job of both capturing the free tropospheric dynamics of the wave and representing its moist convective processes.

d. The coupling between the wave and convection

The simulations presented here are unrealistic in the sense that the interaction between the wave circulation and the convection is one way; that is, a complete picture of the dynamics of 2-day waves will be obtained only once the feedback of the wave circulation on the convective heating is quantified as well. We are currently studying this problem and discuss only preliminary results in this paper. Two points worthy of mention are 1) the rain rate (Fig. 2b) is highly correlated with the column-integrated moist static energy and 2) changes to the latter are dominated by the horizontal divergence associated with the second vertical mode. In particular, on the leading edge of the wave (e.g., near x = −800 km in Fig. 10c), the second vertical mode converges near-surface air with high moist static energy, diverges midlevel air with low moist static energy, and converges upper-tropospheric air with high moist static energy. Perturbations to the moist static energy profile, which closely follow perturbations to the specific humidity profile (Fig. 4b), are most significant in the lower troposphere.

We are exploring the idea that much of the change in the rain rate can be attributed to the change in the gross moist static stability (Neelin and Held 1987) of the first vertical mode. In other words, when the lower troposphere is moistened, the low-level convergence and upper-level divergence associated with the first vertical mode becomes less efficient at “cooling” the column, a localized positive temperature perturbation develops, and the convergence and divergence increase in magnitude in response to the associated changes in the Laplacian of the hydrostatic pressure. Since the divergence circulation of the first vertical mode dominates the moisture budget, the rain rate increases as well. Of course, this theory assumes that the moist static energy converged by the first vertical mode is quickly converted to heat, and much of that converged by the second vertical mode is stored as a water vapor perturbation. While this theory is generally consistent with the observations presented here, a careful analysis of the moist static energy budget of 2-day waves is necessary to test its validity.