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    Longitude–time evolution of 3-hourly satellite-estimated rainfall (averaged over 2.5°–7.5°N) over the central and eastern Pacific during a portion of the 2002 El Niño. Dashed line denotes a westward propagation speed of 18 m s−1.

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    Averaged power spectrum of TRMM rainfall for latitudes in the range 15°S to 15°N. (a) The raw spectrum; (b) the effects of normalizing by a smoothed “red noise” background (see text for details). Solid curves denote the dispersion curves of equatorially trapped shallow-water modes with equivalent depth h = 30 m (pure gravity wave speed of ~17 m s−1). Dashed lines denote the filter boundary for westward-propagating inertia–gravity (WIG) waves. Shading in (b) starts at the 90% confidence interval (1.03) with contour intervals of 0.03.

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    (a) Averaged zonal wavenumber spectrum of TRMM rainfall at the diurnal harmonic (solid) vs a smoothed background (dotted). (b) Averaged zonal distributions of westward- vs eastward-propagating diurnal rain variance (solid vs dotted, respectively) summed over zonal wavenumbers k = 18–32 (i.e., phase speeds 14–25 m s−1). Results reflect averaged quantities from 15°S to 15°N.

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    Examples of (a) genuine and (b) spurious wave objects seen in the TRMM rain data at 5°N (see text for details). Contours denote the wave-object boundary, while shading denotes the raw rainfall.

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    Geographic distribution of time-mean filtered rainfall (shading) associated with the set of genuine WIG wave objects. Thick dark contours denote the ratio of the time-mean filtered rainfall to the time-mean raw rainfall (spatially smoothed for clarity), with levels of either 6% (solid) or 12% (dashed). Boxed regions labeled NA (for North Africa) and WP (for western Pacific) denote the focus of statistical composites in Figs. 610.

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    Composite evolution of raw TRMM rainfall associated with the set of genuine WIG wave objects over (a) North Africa and (b) the western Pacific. Dark solid contours indicate where the composite rain rate is at a quarter of its maximum value, while sloping dashed lines denote a westward propagation speed of 18 m s−1. The solid curves at the bottom of each panel denote the zonal distribution of rainfall at lag 0 h.

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    (top) Composite time–height evolution of high-pass-filtered (<2.5-day) ERA-Interim temperature (contours) and specific humidity (shading) anomalies associated with the set of genuine WIG wave objects over (a) North Africa and (b) the western Pacific. Contours intervals are 0.03 K, with negative values dotted and the zero contour omitted. Only positive moisture anomalies are shaded with intervals of 0.06 K (in units of moist static energy). (bottom) The local composite evolution of raw TRMM rainfall.

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    As in Fig. 7, but with temperature structures truncated to retain only the first four internal vertical modes of the troposphere. The vertical modes were calculated using the algorithm of Fulton and Schubert (1985) with an upper boundary assumed at pressure p = 100 hPa. The base-state virtual temperature profiles used in these calculations were based on composites of low-pass-filtered (>2.5-day) ERA-Interim data. In both cases, the associated phase speeds of the vertical modes are around 51, 22, 18, and 14 m s−1.

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    Composite low-pass-filtered (>2.5-day) zonal winds for the set of genuine wave objects over North Africa (solid black) and the western Pacific (solid gray). Results were obtained as averages over lags −12 to +12 h. Dotted curves denote idealizations of the observed profiles, used in numerical simulations described in section 4.

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    Composite high-pass-filtered (<2.5-day) geopotential height anomalies (shading plus thin contours) and wind vectors on the 1000-hPa surface at lag 0 h for the set of genuine wave objects over (a) North Africa and (b) the western Pacific. Dark (light) shading denotes positive (negative) geopotential anomalies with amplitude > 2 m2 s−2; thin contour interval is 2 m2 s−2. Thick contours denote the composite raw TRMM rainfall with intervals of 1 mm h−1.

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    Composite analysis of 28 squall lines documented over Niamey, Niger, during AMMA by Rickenbach et al. (2009). (a) Longitude–time evolution of raw TRMM rainfall (shading), together with filtered rainfall anomalies associated with WIG wave objects (contours). The sloping dashed line denotes a phase speed of 16.5 m s−1. (b) Time–height evolution of high-pass-filtered (<2.5-day) temperature (contours) and specific humidity (shading) anomalies. Contour intervals are 0.1 K, with negative values dotted and the zero contour omitted. Only positive moisture anomalies are indicated with shading intervals of 0.2 K (in units of moist static energy). See text for details.

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    TRMM rainfall evolution (shading) during a 12-day period of the TRMM–LBA field program, together with filtered rainfall anomalies (contours) associated with WIG wave objects. The dashed lines denote a westward propagation speed of 18 m s−1 and highlight the 18 February squall line studied by Rickenbach (2004).

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    (a) Meridional profile of imposed zonally uniform SSTs, along with (b) the nested grid configuration, used in idealized WRF Model simulations of convection on an equatorial beta plane.

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    Longitude–time evolution of the simulated surface rain (averaged between 5°S and 5°N) during the first 30 days of (a) the control run (U = 0 at all levels) and (b) the sheared run UNA. See text for details.

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    Averaged space–time spectra of the simulated rain during days 15–45 of the runs: (a) control, (b) UNA, (c) UWP, and (d) U5E (see text for details). Shading levels increase logarithmically, starting at 3.5 mm2 day−2. The quantity RW/E denotes the ratio of total westward- to eastward-propagating rain variance. The quantity cp denotes the phase speed associated with the maximum in westward-moving variance (diamonds), except in (d), where the maximum in eastward-moving variance is indicated. Solid curves denote the dispersion curves of Kelvin and n = 1 inertia–gravity waves with equivalent depth h = 37 m. Dotted curves in (d) are similar but include the effects of Doppler shifting by mean easterlies of 5 m s−1. The dotted region in (a) denotes the boundaries of a filter used to isolate WIG wave signals. Power associated with fluctuations in the zonal mean (i.e., k = 0) has been suppressed for clarity. The spectra were obtained as averages over latitudes in the range 2.5°S–2.5°N. The maximum resolved wave period is 8 days.

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    Composite structures of the simulated WIG waves produced in the control run. (a) 10-m horizontal wind vectors, together with surface rainfall (shading) and pressure anomalies (contours) at lag 0 h. Shading intervals are 0.5 mm h−1, starting at 0.5 mm h−1. Contour intervals are 0.1 hPa, with negative values dashed and the zero contour omitted. (b) (top) Time–height evolution of temperature (contours) and specific humidity (shading) anomalies at the equator with (bottom) surface rainfall. Plotting convention is as in Fig. 11b. Results were obtained through lagged linear regression of raw data onto WIG-filtered rainfall time series for a set of 40 evenly spaced base points on the equator—the regression coefficients were averaged over the 40 base points. The filtered rainfall time series at each base point were normalized to have a standard deviation of unity.

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    (a) Schematic of various geometric properties associated with an individual wave object; shading denotes WIG-filtered rainfall associated with the object. (b) Similar to (a), but for an analytic (rectangular-cosine) function F centered on the wave object’s center of mass (white diamonds). See text for details.

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    Correlation between F and the raw (dotted) vs wave-object (solid) rain field as a function of orientation angle for the (a) genuine and (b) spurious wave objects in Fig. 4. The solid horizontal line denotes the threshold for statistical significance of the assigned propagation speed cr at the 95% confidence level.

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Squall Lines and Convectively Coupled Gravity Waves in the Tropics: Why Do Most Cloud Systems Propagate Westward?

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  • 1 CIRES, University of Colorado, and NOAA/Earth System Research Laboratory, Boulder, Colorado
  • 2 NOAA/Earth System Research Laboratory, Boulder, Colorado
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Abstract

The coupling between tropical convection and zonally propagating gravity waves is assessed through Fourier analysis of high-resolution (3-hourly, 0.5°) satellite rainfall data. Results show the familiar enhancement in power along the dispersion curves of equatorially trapped inertia–gravity waves with implied equivalent depths in the range 15–40 m (i.e., pure gravity wave speeds in the range 12–20 m s−1). Here, such wave signals are seen to extend all the way down to zonal wavelengths of around 500 km and periods of around 8 h, suggesting that convection–wave coupling may be important even in the context of mesoscale squall lines. This idea is supported by an objective wave-tracking algorithm, which shows that many previously studied squall lines, in addition to “2-day waves,” can be classified as convectively coupled inertia–gravity waves with the dispersion properties of shallow-water gravity waves. Most of these disturbances propagate westward at speeds faster than the background flow. To understand why, the Weather Research and Forecast (WRF) Model is used to perform some near-cloud-resolving simulations of convection on an equatorial beta plane. Results indicate that low-level easterly shear of the background zonal flow, as opposed to steering by any mean flow, is essential for explaining the observed westward-propagation bias.

Corresponding author address: Stefan Tulich, CIRES-NOAA/ESRL R/PSD1, 325 Broadway, Boulder, CO 80305-3328. E-mail: stefan.tulich@noaa.gov

Abstract

The coupling between tropical convection and zonally propagating gravity waves is assessed through Fourier analysis of high-resolution (3-hourly, 0.5°) satellite rainfall data. Results show the familiar enhancement in power along the dispersion curves of equatorially trapped inertia–gravity waves with implied equivalent depths in the range 15–40 m (i.e., pure gravity wave speeds in the range 12–20 m s−1). Here, such wave signals are seen to extend all the way down to zonal wavelengths of around 500 km and periods of around 8 h, suggesting that convection–wave coupling may be important even in the context of mesoscale squall lines. This idea is supported by an objective wave-tracking algorithm, which shows that many previously studied squall lines, in addition to “2-day waves,” can be classified as convectively coupled inertia–gravity waves with the dispersion properties of shallow-water gravity waves. Most of these disturbances propagate westward at speeds faster than the background flow. To understand why, the Weather Research and Forecast (WRF) Model is used to perform some near-cloud-resolving simulations of convection on an equatorial beta plane. Results indicate that low-level easterly shear of the background zonal flow, as opposed to steering by any mean flow, is essential for explaining the observed westward-propagation bias.

Corresponding author address: Stefan Tulich, CIRES-NOAA/ESRL R/PSD1, 325 Broadway, Boulder, CO 80305-3328. E-mail: stefan.tulich@noaa.gov

1. Introduction

Tropical deep convection is known to interact strongly with atmospheric wave circulations. Well-known examples from the realm of synoptic meteorology include easterly waves, convectively coupled Kelvin waves, and the Madden–Julian oscillation (MJO), all of which have characteristic zonal scales in terms of cloudiness and precipitation of thousands of kilometers [see Kiladis et al. (2009) for a review]. These synoptic-to-intraseasonal phenomena are not only of scientific interest but also have societal impacts as a source of week-2 predictability and beyond (Wheeler and Weickmann 2001; Miura et al. 2007; Mapes et al. 2008).

In addition to synoptic-scale waves, however, there is observational evidence, mainly in the form of case studies, of a myriad of tropical wave–like disturbances with horizontal scales typically measured in hundreds, rather than thousands, of kilometers. The corresponding period of these features is generally less than 2.5 days, with both diurnal and nondiurnal examples appearing prominently. Satellite observations in Fig. 1, for example, reveal a plethora of high-frequency westward- and eastward-moving disturbances over the central Pacific during the 2002 El Niño. Fast westward-moving disturbances with periods of 1–3 days (sometimes called “2-day waves”) have also been detected over the tropical western Pacific (Nakazawa 1988; Clayson et al. 2002), some of which have been identified as inertia–gravity waves (Takayabu 1994; Chen et al. 1996; Haertel and Johnson 1998; Haertel and Kiladis 2004). Such fast-moving waves are often embedded in larger-scale zonally propagating “envelopes” of convection, resulting in complex, multiscale wave hierarchies (Nakazawa 1988; Chen et al. 1996; Straub and Kiladis 2002; Ichikawa and Yasunari 2007; Tulich and Mapes 2008; Dias et al. 2012).

Fig. 1.
Fig. 1.

Longitude–time evolution of 3-hourly satellite-estimated rainfall (averaged over 2.5°–7.5°N) over the central and eastern Pacific during a portion of the 2002 El Niño. Dashed line denotes a westward propagation speed of 18 m s−1.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Diurnally forced examples include convection near coasts or other landscape forcings (Garreaud and Wallace 1997; Yang and Slingo 2001). For example, Greco et al. (1990) and Garstang et al. (1994) documented nocturnally repeating cloud systems, termed “Amazonian squall lines,” propagating westward across the Amazon basin from the northern coast of Brazil (see also Cohen et al. 1995; Warner et al. 2003; Rickenbach 2004). Such systems are reminiscent of the fast, westward-moving squall lines initiated over the northern highlands of Africa (Laing et al. 2008; Rickenbach et al. 2009), as well as the southward-moving systems initiated over the eastern coast of India (Webster et al. 2002; Zuidema 2003). Shorter-lived nocturnal systems have been seen to move northward off the northern coast of Borneo (Houze et al. 1981) and westward off the western coast of Colombia (Mapes et al. 2003). At higher latitudes, Carbone et al. (2002) documented diurnal eastward-moving envelopes of precipitation over the United States in summer, although the role of advection versus wave processes is not as clear in that setting.

Fourier spectral methods have proven useful for identifying and classifying zonally propagating cloud disturbances in the tropics. For example, it is now clear on the basis of spectral filtering that many eastward-moving “superclusters” of convection (e.g., Nakazawa 1988; Dunkerton and Crum 1995) can more definitively be classified as convectively coupled Kelvin waves with the dispersion properties of shallow-water Kelvin modes (Straub and Kiladis 2002). Meanwhile, studies by Wheeler and Kiladis (1999) and Hendon and Wheeler (2008) have shown how the spectral properties of Kelvin waves are distinct from those of the MJO, implying that the dynamics of these two phenomena are fundamentally different.

This study takes advantage of high-resolution satellite data to examine some of the mesoscale aspects of convection–wave coupling in the tropics. Results in sections 2 and 3 show that much of the coherent variability of tropical convection on the meso-alpha scale (200–2000 km) can be attributed to equatorially trapped inertia–gravity waves with implied equivalent depths in the range 15–40 m (i.e., pure gravity wave speeds in the range 12–20 m s−1). This variability includes 2-day waves over the tropical western Pacific, as well as squall lines and diurnally propagating systems over Africa, South America, and elsewhere. To address the question of why most of these disturbances propagate westward, section 4 describes results from some explicit simulations of convection on an equatorial beta plane. The main findings of this paper are summarized and discussed in section 5.

2. Spectral analysis of observations

The space–time variability of tropical convection was assessed using data from the Tropical Rainfall Measuring Mission (TRMM) 3B42 product. As detailed in Huffman et al. (2007), this product consists of 3-hourly surface rainfall estimates archived on a 0.25° × 0.25° latitude–longitude grid extending from 50°S to 50°N. To expedite the analysis and reduce storage requirements, these data were coarse-grained from 0.25° to 0.5° resolution in space.

Fourier transform methods were used to obtain the space–time spectrum of TRMM rainfall for each latitude belt in the range 15°N–15°S. Their average is depicted in Fig. 2a. The methodology was the same as in Wheeler and Kiladis (1999), except that the rainfall time series was divided into segments lasting 32 (rather than 96) days to increase the number of independent spectral estimates to 125 for the 11-yr period 1999–2009. The maximum resolved wave period is therefore 32 days—that is, long enough to capture key synoptic modes of variability (e.g., Kelvin and easterly waves) but short enough to filter out intraseasonal modes such as the MJO.

Fig. 2.
Fig. 2.

Averaged power spectrum of TRMM rainfall for latitudes in the range 15°S to 15°N. (a) The raw spectrum; (b) the effects of normalizing by a smoothed “red noise” background (see text for details). Solid curves denote the dispersion curves of equatorially trapped shallow-water modes with equivalent depth h = 30 m (pure gravity wave speed of ~17 m s−1). Dashed lines denote the filter boundary for westward-propagating inertia–gravity (WIG) waves. Shading in (b) starts at the 90% confidence interval (1.03) with contour intervals of 0.03.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Figure 2b shows the raw spectrum of Fig. 2a normalized by a smoothed “red noise” background. The latter was obtained following Wheeler and Kiladis (1999), except that the smoothing was applied only in the direction of constant frequency, to avoid “bleeding” of power from the diurnal harmonics. The shading in Fig. 2b denotes regions where the signal-to-noise ratio exceeds 1.03, corresponding to the 90% confidence level of statistical significance, based on a chi-square test with 3500 degrees of freedom. The latter number represents a conservative estimate, based on the fact that fluctuations in rainfall cannot be considered as independent of latitude.

The signal-to-noise spectrum of Fig. 2b conveys much of the same information seen in previous studies of the space–time variability of tropical convection (e.g., Wheeler and Kiladis 1999; Roundy and Frank 2004; Cho et al. 2004). In this case, however, the signals of westward-propagating inertia–gravity (WIG) waves are seen to extend continuously all the way out to zonal wavelengths of about 500 km and periods of about 8 h (i.e., well into the meso-alpha scale). A similar statement holds true for eastward-moving synoptic Kelvin waves, whose signals extend out to zonal wavelengths of around 1200 km and periods of around a day, scales that are a factor of 3 smaller than in previous work. Such a broad range of frequencies and wavenumbers implies disturbances that are relatively compact in physical space (cf. Bracewell 2000, p. 160), consistent with the longitude–time evolution of rainfall in Fig. 1.

While Fig. 2 provides clear evidence of WIG waves coupled to convection, the same cannot be said for n = 1 eastward-moving inertia–gravity (EIG) waves, whose signals are not readily apparent. One exception is the distinct peak in Fig. 2b at the eastward-moving diurnal harmonic (near wavenumber k = 22), which is suggestive of a distinct population of diurnal EIG waves (or some combination of EIG and high-frequency Kelvin waves). Evidence for such eastward-moving waves can be found in the recent work of Ichikawa and Yasunari (2007), who observed numerous diurnal cloud systems moving eastward at speeds of 15–20 m s−1 within a single MJO event during its passage over the Maritime Continent. The diurnal cloud systems were found to have zonal wavelengths of about 2000 km, consistent with the location of the eastward-moving diurnal peak in Figs. 2b.

However, such eastward-moving diurnal waves seem to be the exception rather than the general rule. For example, Fig. 3a shows that the global variance of westward-propagating diurnal (WD) rain signals is generally more than 80% larger than the variance of eastward-propagating diurnal (ED) signals for zonal wavenumbers in the range k = 18–32 (i.e., phase speeds in the range 14–26 m s−1). Moreover, looking geographically, Fig. 3b shows that the variance of WD rain signals over Indonesia is around 50% larger than the variance of ED signals (for k = 18–32), while over Africa and South America the bias toward westward propagation is even larger with differences in the range 200%–500%. The explanation for these geographic variations is not clear but may be due in part to variations in the background zonal flow, as illustrated herein.

Fig. 3.
Fig. 3.

(a) Averaged zonal wavenumber spectrum of TRMM rainfall at the diurnal harmonic (solid) vs a smoothed background (dotted). (b) Averaged zonal distributions of westward- vs eastward-propagating diurnal rain variance (solid vs dotted, respectively) summed over zonal wavenumbers k = 18–32 (i.e., phase speeds 14–25 m s−1). Results reflect averaged quantities from 15°S to 15°N.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

In summary, the above results show that moist gravity waves in the tropics tend to move predominantly westward. This bias implies a strong asymmetry in the wave-induced flux of momentum from the troposphere to middle atmosphere, which could ultimately impact the quasi-biennial oscillation of the zonal-mean flow in the tropical stratosphere (Kawatani et al. 2010; Ortland et al. 2011). While the westward bias is likely due in part to advection by mean easterlies, there is also a large body of theoretical and numerical modeling work to suggest an important role of vertical shear (Lindzen and Tung 1976; Rotunno et al. 1988; Silva Dias and Ferreira 1992; Zhang and Geller 1994; Liu and Moncrieff 2001; Stechmann and Majda 2009). Alternatively, studies by Yoshizaki (1991) and Liu et al. (2011) have argued that a westward propagation bias should tend to develop “spontaneously,” even in the absence of a background flow, owing to the effects of planetary rotation [see also Fig. 4 of Andersen and Kuang (2008)]. Addressing these issues is a primary goal of this study.

3. Structures and climatology of observed WIG waves

Leaving aside for the moment questions about what causes the westward-propagation bias of moist gravity waves in the tropics, this section seeks to show that such waves encompass a broad family of disturbances, ranging from 2-day waves over the open ocean to squall lines and diurnally propagating systems over land. To achieve this objective, we developed a general method for tracking coherent wavelike disturbances in satellite convection proxy data. The following paragraphs provide a brief description of the method, given in the context of the TRMM rain product. Results of implementing the method are described thereafter.

a. Method of wave-object identification

The method employs spectral filtering of the TRMM data to isolate zonally propagating wave signals in the longitude–time domain. Here, the filter was designed to capture WIG wave signals with equivalent depths between 17 and 75 m and periods longer than 8 h, based on the broad spectral peak in Fig. 2 (see the dashed curves). Visual inspection of the filtered data leads to the concept of a “wave object,” defined as a contiguous region in the longitude–time domain where the filtered data exceeds a threshold [see Dias et al. (2012) for further review and applications]. The choice of threshold is somewhat arbitrary, although results were found to be largely insensitive to a range of different values. Here, a value was chosen based on the 2nd percentile of the filtered data for latitudes in the range 5°N–5°S. Choosing such a small percentile (i.e., large threshold) ensures that individual wave objects tend to cover only a small fraction of the domain and thus are more likely to capture genuine signal propagation in the raw field (e.g., Fig. 4a), as opposed to just noise. Even so, Fig. 4b illustrates how seemingly random convective events can sometimes project strongly onto the filter, even though westward propagation in the raw field is not apparent.

Fig. 4.
Fig. 4.

Examples of (a) genuine and (b) spurious wave objects seen in the TRMM rain data at 5°N (see text for details). Contours denote the wave-object boundary, while shading denotes the raw rainfall.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

To eliminate these “spurious” wave objects, we devised a procedure for determining whether the filtered data are capturing genuine propagation in the raw field. Briefly, the procedure involves assigning a dominant propagation speed to each wave object, denoted cw, as well as to the raw field underlying each wave object, denoted cr (see the appendix for further details). “Genuine” wave objects are then defined as objects that satisfy two criteria: 1) statistical confidence in the assigned raw speed is above the 95% level, and 2) the absolute difference between cw and cr is less than 10 m s−1 (with both cr and cw in the range 5–40 m s−1). The remaining set of nongenuine wave objects, representing more than 65% of the population, is defined as spurious.

b. Spatial climatology of WIG waves

The spatial climatology of WIG waves was assessed by time averaging the filtered rainfall associated with the set of genuine wave objects. As shown in Fig. 5, the waves occur throughout much of the tropics with activity being generally enhanced to the immediate west of significant topography and/or land–sea contrasts (e.g., the sharp local maximum off the western coast of Africa near Cameroon at 4°N). The implication is that diurnal forcing of convection and/or circulation provides an effective trigger of the waves, which then amplify as they move westward. However, Fig. 5 also shows a broad area of enhanced wave activity over the central and western North Pacific, which is far removed from the effects of any land forcing [seen also by Kiladis et al. (2009)]. One possible explanation for the latter enhancement is that it simply reflects enhanced amounts of convection in the time mean. Indeed, normalizing the shaded field in Fig. 5 by the time average of the unfiltered TRMM rainfall shows that the waves in this case are most active over the Sahel region of western North Africa, as indicated by the thick dashed contours. Still, the normalization does not completely eliminate the enhanced loading over the Pacific, as indicated by the thick solid contours, so some other factors must also be involved.

Fig. 5.
Fig. 5.

Geographic distribution of time-mean filtered rainfall (shading) associated with the set of genuine WIG wave objects. Thick dark contours denote the ratio of the time-mean filtered rainfall to the time-mean raw rainfall (spatially smoothed for clarity), with levels of either 6% (solid) or 12% (dashed). Boxed regions labeled NA (for North Africa) and WP (for western Pacific) denote the focus of statistical composites in Figs. 610.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

c. Composite wave evolution and structures

Statistical composites were constructed to gain insights into the factors that influence the climatology of WIG waves, as well as to compare the structures of the waves appearing over the open ocean to those appearing over land. In the paragraphs below, we compare composites of genuine wave objects whose center of mass was located over either the western North Pacific (3°–6°N, 153.5°–173.5°E) or western North Africa (10°–13°N, 10.5°W–9.5°E), as indicated by the boxed regions labeled WP and NA in Fig. 5, respectively. Data from more than 1000 wave objects were used to generate each of these composites. The wave object’s center of mass in the longitude–time plane was used as the composite base point.

1) Longitude–time evolution of rainfall

The composite evolution of the raw (unfiltered) rainfall appears broadly similar in the two regions, as illustrated in Figs. 6a and 6b. Both composites show a mesoscale rain envelope moving westward at roughly 18 m s−1 with a lifetime of around 40 h. The zonal width of the African envelope is estimated to be around 400 km, based on the zonal distance at which rainfall drops below a quarter of its maximum value at lag 0 h (see the solid curve at the bottom of Fig. 6a). The Pacific envelope is estimated to be roughly a factor of 2 larger but is still within the meso-alpha range of 200–2000 km.

Fig. 6.
Fig. 6.

Composite evolution of raw TRMM rainfall associated with the set of genuine WIG wave objects over (a) North Africa and (b) the western Pacific. Dark solid contours indicate where the composite rain rate is at a quarter of its maximum value, while sloping dashed lines denote a westward propagation speed of 18 m s−1. The solid curves at the bottom of each panel denote the zonal distribution of rainfall at lag 0 h.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

2) Vertical dynamical structures

Structures of the composite waves were assessed using data from the interim European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-Interim; Simmons et al. 2007). These data are available at 6-hourly intervals on a regular N128 Gaussian grid (nominally ~0.7° resolution) with 37 pressure levels in the vertical. To isolate signals associated with the composite waves, temporal filtering was applied at each grid point to remove temporal fluctuations with periods longer than 2.5 days. Both the first and second harmonics of the diurnal cycle were also removed to exclude fluctuations associated with diurnal cycle. To assign a level of statistical significance to the composite dynamical fields, a Student’s t test was used for the difference of means between populations P1 and P2, where P1 denotes the set of genuine wave objects and P2 denotes the same population but with dates of occurrence (i.e., month, day, year) chosen at random.

Figure 7 depicts the time–height evolution of temperature and specific humidity anomalies associated with the composite waves. Both panels show a rapid buildup of moisture in the lower free troposphere (950–600 hPa) during the 12 h prior to peak rainfall. This buildup is accompanied by strong cooling in the lower half of the troposphere and moderate warming in the upper half, similar to previous observations of tropical squall lines and 2-day waves (Gamache and Houze 1985; Haertel and Johnson 1998) The strong cooling near the surface (1000–950 hPa) can presumably be attributed to convectively generated cold pools. Meanwhile, the tropospheric dipole (cool at 950–500 hPa; warm at 500–200 hPa) is suggestive of a propagating wavelike disturbance with just a few wavelengths of vertical structure. This idea is supported by a vertical mode decomposition of the composite temperature structures of the waves, using the method of Fulton and Schubert (1985). In particular, Fig. 8 shows that the dipole temperature pattern can qualitatively be captured by truncating vertical structure to retain only the first four vertical modes of the troposphere, similar to previous observations by Haertel et al. (2008).

Fig. 7.
Fig. 7.

(top) Composite time–height evolution of high-pass-filtered (<2.5-day) ERA-Interim temperature (contours) and specific humidity (shading) anomalies associated with the set of genuine WIG wave objects over (a) North Africa and (b) the western Pacific. Contours intervals are 0.03 K, with negative values dotted and the zero contour omitted. Only positive moisture anomalies are shaded with intervals of 0.06 K (in units of moist static energy). (bottom) The local composite evolution of raw TRMM rainfall.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Fig. 8.
Fig. 8.

As in Fig. 7, but with temperature structures truncated to retain only the first four internal vertical modes of the troposphere. The vertical modes were calculated using the algorithm of Fulton and Schubert (1985) with an upper boundary assumed at pressure p = 100 hPa. The base-state virtual temperature profiles used in these calculations were based on composites of low-pass-filtered (>2.5-day) ERA-Interim data. In both cases, the associated phase speeds of the vertical modes are around 51, 22, 18, and 14 m s−1.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Taking moisture as a proxy for cloudiness, the tilting patterns of specific humidity in Fig. 7 imply a progression in cloudiness from shallow to deep convection to stratiform anvils, similar to an individual squall line (e.g., Zipser 1969; Houze 1977; Cetrone and Houze 2011). This progression has also been observed in much larger-scale wave phenomena including convectively coupled Kelvin waves and the MJO (Straub and Kiladis 2003; Lin and Johnson 1996; Mapes et al. 2006). Compared to the Pacific composite, the tilts in moisture (and temperature) are not as smooth in the African composite, implying a more discrete transition from shallow to deep convection. The reason for this difference is not clear but may be due to differences in the background zonal flow and/or stability profile. Differences in the background zonal flow are illustrated in Fig. 9, which shows that while both the African and Pacific waves occur in the presence of low-level easterly shear, the amplitude of the shear is significantly larger in the African case, owing to the presence of the African easterly jet.

Fig. 9.
Fig. 9.

Composite low-pass-filtered (>2.5-day) zonal winds for the set of genuine wave objects over North Africa (solid black) and the western Pacific (solid gray). Results were obtained as averages over lags −12 to +12 h. Dotted curves denote idealizations of the observed profiles, used in numerical simulations described in section 4.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

3) Horizontal dynamical structures

Differences are also seen when comparing the horizontal dynamical structures of the waves. Figure 10 compares the distribution of 1000-hPa horizontal winds and geopotential height anomalies associated with the composite waves at lag t = 0 h. Structures in the Pacific composite resemble theoretical predictions for the n = 1 WIG wave [e.g., Fig. 4b of Matsuno (1966)], with perturbations that are roughly symmetric about a relative latitude of −1.0°. In contrast, the African composite displays a more asymmetric structure in the north–south direction, with the largest perturbations occurring to the northeast of the composite base point, similar to observations of West African squall lines (Tetzlaff and Peters 1988). This greater asymmetry is likely due to enhanced coupling between convection and planetary rotation, owing to the disturbance being located well off the equator. Indeed, qualitatively similar asymmetries are seen when looking at composites of off-equatorial WIG waves over other parts of the globe, such as the eastern and western North Pacific (results not shown).

Fig. 10.
Fig. 10.

Composite high-pass-filtered (<2.5-day) geopotential height anomalies (shading plus thin contours) and wind vectors on the 1000-hPa surface at lag 0 h for the set of genuine wave objects over (a) North Africa and (b) the western Pacific. Dark (light) shading denotes positive (negative) geopotential anomalies with amplitude > 2 m2 s−2; thin contour interval is 2 m2 s−2. Thick contours denote the composite raw TRMM rainfall with intervals of 1 mm h−1.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

4) Relation to observed squall lines

The African waves documented here are in many ways similar to the more conventionally defined West African squall lines (e.g., Cetrone and Houze 2011, and references therein). To provide more direct evidence of a relationship, we appeal to ground-based weather observations collected over Niamey, Niger, during the summer of 2006 as part of the African Monsoon Multidisciplinary Analyses (AMMA) field program (Redelsperger et al. 2006). In particular, Rickenbach et al. (2009) used a C-band scanning weather radar to document the passage of 28 mesoscale squall lines over Niamey during 5 July–27 September 2006. Figure 11a shows the composite evolution of TRMM rainfall associated with these 28 squall lines, based on the time series of area-averaged rainfall derived from the Niamey-based radar (data courtesy of Prof. Rickenbach). Lag 0 h corresponds to the time of maximum radar-derived rainfall for periods when a mesoscale squall line was deemed to be present in the radar scan volume. The composite index is therefore independent of our TRMM-based analysis. Nevertheless, Fig. 11a shows a composite evolution in terms of raw TRMM rainfall (shading) that is very similar to Fig. 6a’s composite evolution of wave objects over northern Africa. Indeed the contours in Fig. 11a show that the composite evolution is almost perfectly aligned with our WIG wave object filter. This correspondence is found to hold true in most individual cases (not shown), including the 11 August squall line, studied extensively by Chong (2010).

Fig. 11.
Fig. 11.

Composite analysis of 28 squall lines documented over Niamey, Niger, during AMMA by Rickenbach et al. (2009). (a) Longitude–time evolution of raw TRMM rainfall (shading), together with filtered rainfall anomalies associated with WIG wave objects (contours). The sloping dashed line denotes a phase speed of 16.5 m s−1. (b) Time–height evolution of high-pass-filtered (<2.5-day) temperature (contours) and specific humidity (shading) anomalies. Contour intervals are 0.1 K, with negative values dotted and the zero contour omitted. Only positive moisture anomalies are indicated with shading intervals of 0.2 K (in units of moist static energy). See text for details.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Vertical soundings of temperature and moisture were also obtained at Niamey as part of the AMMA field campaign. These data were typically recorded at 6-h intervals and are available at 5-mb resolution in the vertical, allowing for a detailed view of the vertical dynamical evolution of squall lines over northern Africa. The composite evolution in Fig. 11b is once again very similar to that obtained for WIG waves using the ERA-Interim data (see Fig. 7a). One notable exception, however, is the slightly greater extension of cold air out ahead of the surface-based cold pool in the lower free troposphere between roughly 900 and 600 mb. This greater extension indicates that cooling of the lower free troposphere, rather than cooling of the planetary boundary layer, may be essential for causing the subsequent deepening of convection between lags −6 and 0 h, consistent with Tulich and Mapes (2010).

In addition to squall lines over Africa, numerous squall-type convective systems were sampled over the Amazon region of South America as part of the TRMM Large-Scale Biosphere–Atmosphere (LBA) field program during the early wet season of 1999 (Silva Dias et al. 2002). One of the largest of these systems occurred on 18 February and was classified by Rickenbach (2004) as a coastally generated squall line, of the type previously documented in Cohen et al. (1995). Figure 12 shows that the 18 February squall line can alternatively be classified as a WIG wave, closely matching its westward propagation speed of around 18 m s−1. This system can be tracked for roughly 3 days and travels a zonal distance of more than 3000 km. Other examples of apparent WIG waves can also be seen in Fig. 12, based on filtered data aligning with the coherent disturbances in the raw rain field. In summary, it seems clear that many mesoscale features previously identified as “squall lines” in the literature are identical to what we refer to as convectively coupled inertia–gravity waves.

Fig. 12.
Fig. 12.

TRMM rainfall evolution (shading) during a 12-day period of the TRMM–LBA field program, together with filtered rainfall anomalies (contours) associated with WIG wave objects. The dashed lines denote a westward propagation speed of 18 m s−1 and highlight the 18 February squall line studied by Rickenbach (2004).

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

4. Why do most cloud systems propagate westward?

While the above results indicate that many tropical squall lines and 2-day waves are part of a broad family of inertia–gravity wave disturbances, it remains unclear why most of these disturbances propagate westward. To address this issue, we used the Weather Research and Forecast (WRF) Model to perform a series of explicit, nested simulations of convection on an equatorial beta plane. The setup for these simulations is described below, followed by results.

a. Model setup

The model is configured to simulate convection over a time-invariant ocean with zonally uniform sea surface temperatures (SSTs). As shown in Fig. 13a, the SSTs vary in the north–south direction, ranging from 301 K on the equator (y = 0) to 290 K at latitudes poleward of 10°N and 10°S. The goal is to produce a relatively narrow intertropical convergence zone (ITCZ) with deep convection confined to the lowest latitudes, where model resolution is finest through grid nesting. Specifically, three grids are employed with horizontal mesh spacings of 36, 12, and 4 km (denoted grids 1–3, respectively). Unlike traditional nesting, Fig. 13b shows that all three grids have the same physical extent in the east–west direction, so that changes in horizontal resolution occur only in the north–south direction.

Fig. 13.
Fig. 13.

(a) Meridional profile of imposed zonally uniform SSTs, along with (b) the nested grid configuration, used in idealized WRF Model simulations of convection on an equatorial beta plane.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Additional details about the model setup are as follows. The domain is an equatorial beta plane extending roughly 5000 km in the east–west direction and 9900 km in the north–south direction, with periodic east–west boundary conditions and rigid walls assumed at the northern and southern boundaries. In the vertical, all three grids extend to a height z = 28 km, with 50-m spacing near the surface increasing to 500 m at and above z ≈ 8 km. Convection is maintained though horizontally uniform radiative-like cooling of 1.25 K day−1 between the surface and pressure p = 200 hPa, decaying linearly to zero at p = 150 hPa. Cloud microphysics are parameterized using the single-moment six-class scheme of Hong and Lim (2006). Horizontal mixing by unresolved turbulent eddies is parameterized using a Smagorinsky first-order closure, while vertical mixing is parameterized using the planetary boundary layer scheme of Hong et al. (2006). To remove energy from gravity waves reflected off the upper boundary, Rayleigh damping is applied in the uppermost 7 km of the model domain. Rayleigh-type damping is also applied at all levels in grid columns near the northern and southern boundaries of grid 1.

b. Hypothesis testing

On the basis of the background wind profiles in Fig. 9, we hypothesize that vertical shear of the background zonal flow is essential for producing a westward-propagation bias in high-frequency convective wave activity. To test this idea, we modified the model’s horizontal momentum equations so that the Coriolis force acts only on perturbation winds about the zonal mean at each time step. Additionally, a nudging term of the form was added to the zonal momentum equation, where denotes the zonal-mean zonal wind at each time step, U is a specified zonal wind profile, and τ = 1 h is a relaxation time. These two changes together allowed for direct control of the simulated background u-wind profile, so that the effects of advection by vertically sheared winds could be cleanly tested.

Several simulations were performed, each lasting a period of 45 days, but differing in terms of the specified U-wind profile. The “control” run features no mean wind (i.e., U = 0 at all levels). In this case, Fig. 14a shows that a quasi-standing-wave pattern develops in the simulated rain after a period of roughly 15 days. Spectral analysis of this standing wave pattern (see Fig. 15a) shows the presence of both westward- and eastward-moving inertia–gravity waves with preferred zonal wavelengths of around 1700 km (i.e., k = 3). The waves have implied equivalent depths of around 37 m, corresponding to pure gravity wave speeds of 19 m s−1, similar to observations. Horizontal and vertical structures of the simulated waves, depicted in Fig. 16, are also found to broadly resemble observations, including the horizontal V-shaped pattern in surface rainfall, which is reminiscent of Takayabu (1994)’s observations of convection patterns associated with WIG waves over the equatorial western Pacific.

Fig. 14.
Fig. 14.

Longitude–time evolution of the simulated surface rain (averaged between 5°S and 5°N) during the first 30 days of (a) the control run (U = 0 at all levels) and (b) the sheared run UNA. See text for details.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Fig. 15.
Fig. 15.

Averaged space–time spectra of the simulated rain during days 15–45 of the runs: (a) control, (b) UNA, (c) UWP, and (d) U5E (see text for details). Shading levels increase logarithmically, starting at 3.5 mm2 day−2. The quantity RW/E denotes the ratio of total westward- to eastward-propagating rain variance. The quantity cp denotes the phase speed associated with the maximum in westward-moving variance (diamonds), except in (d), where the maximum in eastward-moving variance is indicated. Solid curves denote the dispersion curves of Kelvin and n = 1 inertia–gravity waves with equivalent depth h = 37 m. Dotted curves in (d) are similar but include the effects of Doppler shifting by mean easterlies of 5 m s−1. The dotted region in (a) denotes the boundaries of a filter used to isolate WIG wave signals. Power associated with fluctuations in the zonal mean (i.e., k = 0) has been suppressed for clarity. The spectra were obtained as averages over latitudes in the range 2.5°S–2.5°N. The maximum resolved wave period is 8 days.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Fig. 16.
Fig. 16.

Composite structures of the simulated WIG waves produced in the control run. (a) 10-m horizontal wind vectors, together with surface rainfall (shading) and pressure anomalies (contours) at lag 0 h. Shading intervals are 0.5 mm h−1, starting at 0.5 mm h−1. Contour intervals are 0.1 hPa, with negative values dashed and the zero contour omitted. (b) (top) Time–height evolution of temperature (contours) and specific humidity (shading) anomalies at the equator with (bottom) surface rainfall. Plotting convention is as in Fig. 11b. Results were obtained through lagged linear regression of raw data onto WIG-filtered rainfall time series for a set of 40 evenly spaced base points on the equator—the regression coefficients were averaged over the 40 base points. The filtered rainfall time series at each base point were normalized to have a standard deviation of unity.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

The evolution of convection in the model is strikingly different in the presence of shear. Simulations were performed with U given by the observed background wind profiles in Fig. 9. In the case of the North African profile (denoted UNA), Fig. 14b shows that westward-moving waves are now strongly preferred, supporting the notion that background flow effects are essential for producing the observed westward-propagation bias. This bias is further illustrated in the rain spectrum of Fig. 15b, which shows a dominance of a WIG wave signals with characteristic speeds of around 26 m s−1 (i.e., 7 m s−1 faster than in the control run). The increase in speed is likely due to the effects of Doppler shifting by mean easterlies through the troposphere; the mass-weighted tropospheric average of UNA is −7.3 m s−1.

Broadly similar results are obtained using the observed western Pacific profile (denoted UWP), as shown in Fig. 15c. The westward bias is slightly weaker, as indicated by the ratio of total westward- to eastward-propagating rain variance RW/E (given in each figure panel). This weakening of the bias can be attributed to the more moderate background shear, together with the effects of mean surface easterlies, via the mechanism of wind-induced surface heat exchange (WISHE; cf. Yano and Emanuel 1991). To isolate the effects of WISHE, an additional run was performed with U = −5 m s−1 at all levels, denoted U5E. Figure 15d shows that the waves in this case are biased toward eastward propagation (RW/E = 0.5), similar to previous two-dimensional (2D) modeling work by Grabowski and Moncrieff (2001) and Liu and Moncrieff (2004).

Simulations were also performed with U given by idealized versions of the composite background wind profiles in Fig. 9, indicated by the dotted curves. Both profiles are characterized by uniform easterly shear at low levels and zero shear aloft (nominally above 5 km). Despite the lack of upper-level shear, both profiles gave a clear bias toward westward propagation (RW/E > 2 in both cases; spectra not shown), indicating that low-level shear is sufficient for explaining the observed westward-propagation bias.

5. Summary and discussion

This study focused on convectively coupled inertia–gravity waves, which are a prominent class of zonally propagating mesoscale cloud disturbances. Results showed that the waves typically move westward at speeds of around 18 m s−1 and are common throughout the tropics, especially over Africa, South America, and the central and western Pacific. Horizontal dynamical structures of the waves were seen to generally resemble theoretical predictions for shallow-water inertia–gravity waves, except over northern Africa, where significant north–south asymmetries were seen in the wind and mass fields. In the vertical, temperature anomalies of the waves were found to be dominated by a dipole pattern in the troposphere, with convection being most active when the lower free troposphere is anomalously cool and moist, ahead of the surface-based cold pool. Many of these results are in agreement with previous studies of “2-day waves” over the tropical western Pacific (e.g., Takayabu et al. 1996; Haertel and Johnson 1998). However, here it was shown that 2-day waves are just one of several types of convectively coupled inertia gravity waves (see also Haertel and Kiladis 2004). Another type was shown to include many tropical squall line systems, including a number of well-documented systems over Africa and South America. To address the question of why most of these disturbances propagate westward, a nested-channel version of the WRF Model was used to perform some near-cloud-resolving simulations of convection on an equatorial beta plane. Results showed that easterly shear of the background zonal wind at low levels was sufficient for producing a westward-propagation bias in the model.

A fundamental question that has long surrounded moist tropical waves is what determines their horizontal speed of propagation. Based on the observed dipole temperature structures of the waves found here and elsewhere, it would seem that waves with relatively short tropospheric vertical wavelengths are important, as opposed to deeper waves feeling the effects of a reduced static stability (cf. Emanuel et al. 1994). This idea is supported by previous analyses of 2D cloud-resolving models, which show a strong coupling between convection and gravity waves with vertical wavelengths comparable to the depth of the troposphere (Oouchi 1999; Grabowski et al. 2000; Shige and Satomura 2001; Tulich et al. 2007; Tulich and Mapes 2008; Lane and Zhang 2011). This coupling involves forcing of the waves by diabatic processes in deep convective and stratiform regions, which then trigger new convection via low-level destabilization. Waves that are most effective at triggering new convection are those that produce cooling and moistening in the “effective inhibition layer,” nominally the lowest 4 km (Tulich and Mapes 2010; Kuang 2010).

The impact of vertical shear on convection–wave coupling has been less well studied in an explicit-convection context. Here it was shown that vertical shear can have a strong influence on the direction of convective wave propagation, similar to the effects of WISHE under a mean surface flow. Possible insight into this behavior can be gleaned from recent theoretical work by Stechmann and Majda (2009), who showed that vertical shear can provoke a strong directional bias in the gravity wave response to imposed convective heating. This bias includes differences in temperature and stability about the heating, implying a preferred direction for further convective development. Biases in convective wave development under shear have been demonstrated in a number of linear modeling studies of convective wave instability, including those of Raymond (1983), Nehrkorn (1986), and Silva Dias and Ferreira (1992). A common theme among all of these studies is their view that squall lines can be regarded as a kind of conditional instability of the flow involving interactions between convection, gravity waves, and vertical shear of the background wind.

Studies by Rotunno et al. (1988) and Weisman et al. (1988) have argued that interactions between surface-based cold pools and vertical shear may also be important for the maintenance of long-lived squall lines. This idea is supported by idealized simulations of gravity currents, which show that vertical lifting at the downshear edge of a gravity current is generally enhanced in the presence of low-level vertical shear. In the tropics, however, it seems evident that gravity current speeds are generally too slow to account for the fast-moving squall lines documented here and elsewhere. Moreover, squall lines in both the tropics and midlatitudes are often seen to propagate via discrete “jumps,” which is difficult to explain using gravity current arguments (e.g., Zipser 1977; Houze 1977; Carbone et al. 1990; Grady and Verlinde 1997; Fovell et al. 2006).

In summary, it may be that the most vigorous and long-lived squall lines are ones in which tropospheric gravity waves and surface cold pools have commensurate scales and speeds to provide a resonant forcing of convection. Further observations and simulations will be needed to address this issue more fully in future work.

Acknowledgments

Special thanks go to Prof. Tom Rickenbach, who provided the AMMA radar data, as well as to Dr. Paul Ciesielski, who provided the AMMA sonde data. Instructive discussions with Drs. Stan Trier and Rich Rotunno and Prof. Brian Mapes were helpful in preparing the manuscript. The WRF Model simulations were performed using computing time made available by the Advanced Computing Section of the National Ocean and Atmosphere Administration’s Earth System Research Laboratory and by the Computational and Information Systems Laboratory of the National Center for Atmospheric Research, which is supported by the National Science Foundation. This research was supported by the National Science Foundation under Grant ATM-0806553.

APPENDIX

Assignment of Propagation Speeds cw and cr

The method for assigning the propagation speeds cw and cr involves several steps. First, a wave object’s duration in grid points nt, zonal span nx, area coverage Aw, and center of mass are computed, as illustrated in Fig. A1a. Next, following the approach of Carbone et al. (2002), an analytic function F is defined whose values are constant in one direction, denoted by orientation angle θ, and variable in the direction perpendicular, denoted θ (see Fig. A1b). The space–time region over which F is defined is rectangular in shape with dimensions D and D, where and D ≡ 1.27 Aw/D. The form of F in the direction θ is given by F ≡ cos(πx/D), where x ranges from −D/2 to +D/2.

Fig. A1.
Fig. A1.

(a) Schematic of various geometric properties associated with an individual wave object; shading denotes WIG-filtered rainfall associated with the object. (b) Similar to (a), but for an analytic (rectangular-cosine) function F centered on the wave object’s center of mass (white diamonds). See text for details.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

Rotation of F (about the wave object’s center of mass) is used to obtain values of cw and cr. In particular, we compute the correlation between F and the wave object’s filtered anomaly field for θ ranging between 0° and 90° in 1° increments. The angle yielding the maximum correlation is taken as the wave object speed cw. Similarly, the angle yielding the maximum correlation between F and the underlying raw field is used to assign the raw speed cr. To assign a level of statistical significance to cr, we compute the mean and standard deviation of the correlation between F and the raw field for all θ between 0° and 180°. Only cases where the maximum correlation exceeds the mean correlation by 1.65 standard deviations are taken to be statistically significant at the 95% level. The latter condition is based on a one-sided t test for the difference of means between two populations with different sample sizes but the same standard deviation.

Considering a specific example, Fig. A2a compares the correlation between F and the filtered (solid) versus raw (dotted) rain field as a function of θ for the genuine wave object depicted in Fig. 4a. The two curves appear broadly similar with peaks near θ = 20°, corresponding to phase speeds of around 15 m s−1. The peak correlation between F and the raw field (used to assign cr) is well above the 95% confidence level for statistical significance, indicated by the thin solid line. These results differ from the spurious wave object in Fig. 4b, whose raw correlation is just above the 95% significance level (see Fig. A2b), while cr differs from cw by more than 10 m s−1.

Fig. A2.
Fig. A2.

Correlation between F and the raw (dotted) vs wave-object (solid) rain field as a function of orientation angle for the (a) genuine and (b) spurious wave objects in Fig. 4. The solid horizontal line denotes the threshold for statistical significance of the assigned propagation speed cr at the 95% confidence level.

Citation: Journal of the Atmospheric Sciences 69, 10; 10.1175/JAS-D-11-0297.1

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