Thermodynamic Aspects of Tropical Cyclone Formation

1. Introduction

A preexisting low-level cyclonic disturbance is one of the necessary conditions for tropical cyclone (TC) formation (e.g., Gray 1968) in the present climate. Over the Atlantic basin, a vast majority of tropical cyclones form from tropical easterly waves (e.g., Landsea 1993). Most of these waves originate in Africa, the so-called African easterly waves (e.g., Carlson 1969; Burpee 1972; Thorncroft and Hoskins 1994a,b), and some of them develop in situ over the Atlantic.

It has been an intriguing question how a tropical storm protovortex forms within a synoptic-scale disturbance and survives in the generally hostile tropical environment before evolving into a self-sustaining entity. Recent work by Dunkerton et al. (2009) showed that a protected region of cyclonic recirculation exists in the lower troposphere in some, but not all, easterly waves. Such a region, the so-called wave pouch, straddles the “critical latitude” where the wave phase speed and the mean flow speed are equal. The recirculation occurs about the trough axis inside the wave’s critical layer, which forms from the nonlinear interaction between the wave and the flow around the critical latitude. As demonstrated by both observational diagnoses (Dunkerton et al. 2009; Montgomery et al. 2010a; Wang et al. 2012a; Raymond and López Carrillo 2011) and real-case and idealized numerical model simulations (Wang et al. 2010a,b; Montgomery et al. 2010b; Fang and Zhang 2010; Fritz and Wang 2012, manuscript submitted to J. Atmos. Sci.), a wave pouch can protect a protovortex inside from strain/shear deformation and dry air intrusion above the boundary layer, and its cyclonic gyre also provides a focal point for vorticity aggregation.

Wang et al. (2012b) further examined the vertical structure of the wave pouches during the 2009 Pre-Depression Investigation of Cloud Systems in the Tropics (PREDICT) field experiment dry run over the Atlantic, and suggested that a moist and diabatically activated wave pouch extending from the midtroposphere (600–700 hPa) down to the boundary layer is a necessary and highly favorable condition for tropical cyclone formation. Davis and Ahijevych (2012) examined the mesoscale structural evolution of two developing disturbances and one nondeveloping disturbance surveyed in the PREDICT field experiment. Using geostationary satellite data and multisensor derived precipitation, they showed that deep convection, although characterized by strong diurnal variations prior to genesis, recurred near the pouch center, and they suggested that the wave pouch plays an important role in mesoscale organization.

Tropical cyclone formation is intrinsically a multiscale process. The marsupial paradigm suggests that the meso-α-scale wave pouch provides a localized, favorable environment for convective organization, vorticity aggregation, and tropical cyclone formation. As for the development of the meso-β-scale tropical cyclone protovortex near the surface, there are two groups of theories: top-down development and bottom-up development [see the review paper by Tory and Montgomery (2006) for more details]. In the top-down development, a midlevel vortex, which presumably forms in a stratiform region, can induce the surface cyclonic circulation by building downward (Ritchie and Holland 1997; Simpson et al. 1997; Bister and Emanuel 1997). In the bottom-up development theory, cyclonic vorticity anomalies are generated by condensational heating in the lower troposphere, and an upscale vorticity cascade leads to the formation of a tropical depression vortex (Hendricks et al. 2004; Montgomery et al. 2006). Briefly speaking, the top-down development emphasizes the importance of stratiform processes and a midlevel vortex in initiating the surface cyclonic circulation, while the bottom-up development emphasizes the critical role of deep convection and the associated low-level convergence. The vorticity conservation principle of Haynes and McIntyre (1987) suggests that there is no net vorticity transport across isobaric surfaces, and recent studies have shown that the low-level convergence is the most effective way to intensify the surface circulation (e.g., Tory and Montgomery 2008; Wang et al. 2010a; Montgomery and Smith 2010; Fang and Zhang 2010; Raymond et al. 2011). On the other hand, Raymond et al. (2011) suggested that a midlevel vortex, with its associated cold core in the lower troposphere and warm core in the upper troposphere, can significantly modify the vertical mass flux profile and create a thermodynamically favorable environment for tropical cyclone formation.

In this study, we will address the following two questions:

• How does the circulation evolve at the meso-α (~wave pouch) and meso-β (~protovortex) scales during TC formation?

• What thermodynamic conditions help to make the pouch center a preferred location for recurring convection and tropical cyclone genesis?

The outline of this paper is as follows. Section 2 is a brief description of the model simulation and the dropsonde data from a recent field campaign. The evolution of the meso-α and meso-β circulations and the thermodynamic structure of the wave pouch in a high-resolution model simulation are examined in section 3. In section 4, the Sawyer–Eliassen equation is used to examine the balanced response to convective and stratiform heating derived from the numerical model simulation. Analysis of the dropsonde data is presented in section 5, followed by discussion and concluding remarks in section 6.

2. Model simulation and the PREDICT field data

The high-resolution numerical model simulation of pre-Hurricane Felix (2007) by Wang et al. (2010a,b) is further diagnosed in this study. Hurricane Felix originated from an African easterly wave and developed into a tropical cyclone at 2100 UTC 31 August 2007, over the central Atlantic. The formation of Felix within the wave critical layer and the evolution of convective and stratiform precipitation were examined by Wang et al. (2010a,b) in the numerical simulations using the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2005). The high-resolution numerical simulation examined in the present study employed nested grids with horizontal resolutions of 81, 27, 9, and 3 km. Convection was resolved explicitly in the inner two grids, and the Kain–Fritsch cumulus scheme (Kain and Fritsch 1990) was used in the outer two grids. The model was initialized at 0000 UTC 29 August 2007 and was driven by the European Centre for Medium-Range Weather Forecasts (ECMWF) 6-hourly analyses. A small closed circulation formed at 1000 hPa around t = 40 h (or 1500 UTC 30 August) in the model simulation in the earth-relative frame of reference. To be consistent with forecasters’ criteria for the formation of a tropical depression, we define t = 40 h as the genesis time in this study, although Wang et al. (2010a) argued that a tropical depression may have formed during 30–40 h as indicated by the quasi-Lagrangian flow evolution in the wave’s comoving frame of reference.¹ The storm reaches hurricane strength by the end of the 3-day simulation.

Dropsonde data from the PREDICT field experiment (Montgomery et al. 2012) were diagnosed to verify some conclusions drawn from the numerical model simulation. The PREDICT field experiment was sponsored by the National Science Foundation (NSF) and was carried out over the west Atlantic from 15 August to 30 September 2010. The NSF–National Center for Atmospheric Reseach (NCAR) Gulfstream V (GV) aircrafts were employed in the field experiment. Eight disturbances were surveyed, including four developers and four nondevelopers. The dynamical forecast method based on the marsupial paradigm (Wang et al. 2009) was used to predict the track of possible genesis locations, and flight patterns were designed based on the tracks. More than 500 dropsondes were released from around 40 000 ft (~12 km or 200 hPa), and 558 drops were included in the quality controlled data archive, which provided an ideal dataset to investigate the thermodynamic evolution of the wave pouch prior to genesis.

3. Diagnoses of the high-resolution numerical simulation

a. Evolution of the meso-α and meso-β circulation

A wave pouch is a meso-α scale structure within a synoptic-scale wave, in contrast to a TC protovortex, which is at the meso-β scale. To examine the circulation evolution at different spatial scales, relative vorticity from the WRF Model simulation is averaged over a 2° square box and a 6° square box following the propagating pouch center to represent the meso-β-scale circulation and the meso-α circulation, respectively. The 2° box average of vorticity has been examined by Wang et al. (2010a), and the figure is reproduced here (Fig. 1a) for comparison.

Time–height cross section of relative vorticity averaged in (a) a 2° square box and (b) a 6° square box following the pouch center in the numerical model simulation of Felix. [Figure 1a is reproduced from Wang et al. (2010a).]

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Time–height cross section of relative vorticity averaged in (a) a 2° square box and (b) a 6° square box following the pouch center in the numerical model simulation of Felix. [Figure 1a is reproduced from Wang et al. (2010a).]

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Time–height cross section of relative vorticity averaged in (a) a 2° square box and (b) a 6° square box following the pouch center in the numerical model simulation of Felix. [Figure 1a is reproduced from Wang et al. (2010a).]

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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As described in Wang et al. (2010a), the relative vorticity maximum averaged over the 2° box (the meso-β scale) is initially located between 600 and 700 hPa, which is associated with the precursor easterly wave peaking at the level of the easterly jet. During 22–36 h, there is an apparent downward development of the cyclonic circulation as vorticity increases near the surface. However, the decrease of the midlevel vorticity at the same time suggests that the spinup of the surface circulation is not due to the intensification of the midlevel vortex. This evolution sequence is thus different from what is suggested by the top-down theory (Ritchie and Holland 1997; Simpson et al. 1997). The vorticity budget analysis in Wang et al. (2010a) showed that the vorticity increase near the surface is mainly due to the low-level convergence, consistent with the bottom-up development theory (Hendricks et al. 2004; Montgomery et al. 2006). As the low-level vorticity continues to increase, strong cyclonic vorticity extends upward. A warm-core structure has formed by 36 h at the meso-β scale with maximum vorticity near the top of the boundary layer (~850 hPa), and a troposphere-deep vortex has developed by the end of the simulation.

The time–height evolution of vorticity at the meso-α scale is shown in Fig. 1b. Averaged over a larger area, the vorticity magnitude at the meso-α scale is much smaller than that at the meso-β scale. Stokes’ theorem states that the circulation about a closed loop in a horizontal plane is equal to the integral of the vertical vorticity over the area enclosed by the loop. Comparison of Figs. 1a and 1b suggests that the 2° box near the pouch center accounts for 40%–50% of the circulation about the 6° box at 800 hPa. The relative vorticity averaged over the 6° box initially peaks around 650 hPa, similar to the 2° box average. Different from the 2° box average, the 6° box-averaged vorticity at 650 hPa and below increases monotonically, and the maximum vorticity remains at 650 hPa prior to 42 h, suggesting a cold-core structure in the lower troposphere at the meso-α scale. Because of the rapid increase of vorticity between 750 and 850 hPa, the maximum vorticity shifts to a lower level (750–800 hPa) after 42 h, but it remains at a higher altitude than the maximum vorticity at the meso-β scale even at the hurricane stage.

To understand why the vorticity evolution is different at different spatial scales, we examined the vorticity budget. The vorticity equation in isobaric coordinates can be written as (Haynes and McIntyre 1987; Tory and Montgomery 2008; Raymond et al. 1998; Wang et al. 2010a)

View Expanded

where η is the absolute vorticity, V′ is the wave-relative horizontal flow, p is pressure, ω is the vertical velocity in isobaric coordinates, and is the vertical unit vector. The term on the left-hand side (lhs) of the equation is the local tendency of the absolute vorticity in the wave’s comoving frame of reference; the first term on the right-hand side (rhs) is the convergence of advective vorticity flux, which incorporates the horizontal advection of the absolute vorticity and the stretching effect; the second rhs term is the convergence of the nonadvective vorticity flux, which is the sum of the vertical advection of the vertical vorticity and the tilting effect; and the last term on the rhs is the residual term, including diffusion and subgrid processes. The vorticity budget terms are usually very noisy (Wang et al. 2010a). To get a smooth evolution pattern, we integrated Eq. (1) with time:

View Expanded

where the lhs term represents the net change of absolute vorticity during the time interval t − t₀, and the rhs terms represent the accumulative effects of different processes during the same time period.

Figure 2 shows the net vorticity tendency and the convergence of the advective vorticity flux integrated from 22 h onward [i.e., t₀ is set to 22 h and t varies from 22 to 54 h in Eq. (2)], averaged over a 2° square box (meso-β scale; left panels) and a 6° square box (meso-α scale; right panels), respectively. At the meso-β scale, positive tendency is dominant below 700 hPa throughout the calculation period, indicating persistent spinup of the low-level circulation. In particular, the strong positive tendency after 30 h marks the rapid increase of the low-level cyclonic vorticity. Negative vorticity tendency is present in the upper and middle troposphere at the early stage, consistent with the midlevel spindown at the meso-β scale as shown in Fig. 1a. The negative tendency diminishes in both depth and magnitude after 36 h, but it is not completely replaced by positive tendency until 48 h, after a tropical cyclone has formed.

(a),(c) Absolute vorticity tendency (10⁻⁵ s⁻¹) and (b),(d) the convergence of advective vorticity flux (10⁻⁵ s⁻¹) integrated from t = 22 h (2200 UTC 29 Aug) onward in the numerical model simulation of Felix, for (left) 2° and (right) 6° box average.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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(a),(c) Absolute vorticity tendency (10⁻⁵ s⁻¹) and (b),(d) the convergence of advective vorticity flux (10⁻⁵ s⁻¹) integrated from t = 22 h (2200 UTC 29 Aug) onward in the numerical model simulation of Felix, for (left) 2° and (right) 6° box average.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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(a),(c) Absolute vorticity tendency (10⁻⁵ s⁻¹) and (b),(d) the convergence of advective vorticity flux (10⁻⁵ s⁻¹) integrated from t = 22 h (2200 UTC 29 Aug) onward in the numerical model simulation of Felix, for (left) 2° and (right) 6° box average.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Wang et al. (2010a) showed that the convergence of the advective vorticity flux plays a dominant role in spinning up the low-level circulation. We will thus focus on this forcing term. The advective vorticity flux term is positive below 600 hPa and increases significantly near the surface after 27 h. Above 600 hPa, the advective vorticity flux term contributes to weak negative vorticity tendency. Wang et al. (2010a) showed that the vorticity increase in the middle and upper troposphere is mainly due to the nonadvective vorticity flux term. As illustrated by Tory and Montgomery (2008) and Tory and Frank (2010), the nonadvective vorticity flux term in a warm-core vortex contributes to a positive tendency inward of an updraft but an opposite tendency at larger radii. Therefore, although there is no net vorticity transport across isobaric surfaces, the nonadvective vorticity flux term may induce a localized positive vorticity tendency near the pouch center.

The meso-α-scale circulation demonstrates a different evolution from the meso-β-scale one (Fig. 2c). The advective vorticity flux term still contributes to positive tendency in the lower troposphere (Fig. 2d), but it is much weaker than that at the meso-β scale and is not sufficient to counterbalance the frictional spindown effect in the boundary layer at the early stage. The net vorticity tendency is negative below 900 hPa before 30 h, but the advective vorticity flux term results in a positive net tendency or spinup in the middle troposphere, prior to the surface spinup (Fig. 2c). Again, this should not be interpreted as a “top-down” development as the low-level convergence associated with deep convection, instead of vertical advection, plays the essential role in spinning up the surface circulation at the meso-α scale. Also note that the surface spinup at the meso-β scale precedes the meso-α-scale spinup both near the surface and in the middle troposphere.

The stretching effect is a major player in the advective vorticity flux term. The time–height cross sections of divergence averaged over the 2° square box and the 6° square box are shown in Fig. 3. In both areal averages, strong convergence occurs near the surface, and weak convergence extends up to 400 hPa. The convergence averaged over the 2° square box is comparable to that averaged over the 6° square box in the middle troposphere, but the former is 3–4 times stronger than the latter in the boundary layer. The stronger low-level convergence near the pouch center is associated with the spatial distribution of convective and stratiform precipitation within the wave pouch. Convective precipitation is characterized by low-level convergence and upper-level divergence while stratiform precipitation features midlevel convergence and divergence above and below (e.g., Mapes and Houze 1995). As shown in Fig. 3 of Wang et al. (2010b), stratiform precipitation covers a larger fraction of the wave pouch than convective precipitation, especially at the early stage. Deep convection, although less spatially extensive, is often associated with heavy precipitation rates. It tends to recur near the pouch center and is more transient in the outer pouch region. Both deep convective and stratiform precipitation rates increase with time, but the former increases more than the latter. The mean heating profile and the mean divergence profile thus become increasingly convective near the pouch center, and the associated low-level convergence is sufficiently strong to spin up the surface circulation near the pouch center. Compared to the meso-β scale, stratiform precipitation makes a relatively larger contribution to the total precipitation at the meso-α scale (or over the entire pouch), and the relatively weak low-level convergence is not sufficient to offset the frictional spindown effect at the early stage. The surface spinup at the meso-α scale thus lags the surface spinup at the meso-β scale, and it also lags the midlevel spinup at the meso-α scale.

Time–height cross section of the divergence (10⁻⁵ s⁻¹) averaged in (a) a 2° square box and (b) a 6° square box following the pouch center in the numerical model simulation of Felix.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Time–height cross section of the divergence (10⁻⁵ s⁻¹) averaged in (a) a 2° square box and (b) a 6° square box following the pouch center in the numerical model simulation of Felix.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Time–height cross section of the divergence (10⁻⁵ s⁻¹) averaged in (a) a 2° square box and (b) a 6° square box following the pouch center in the numerical model simulation of Felix.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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The different evolutions of circulation at different spatial scales have implications for the analysis of field data and interpretation of the analysis. Circulation or spatial-averaged vorticity is often calculated to estimate the storm intensity, especially when dropsonde data are used. The above diagnoses suggest that vorticity averaged over a large area (or circulation enclosing a large area) may not represent the development of a TC protovortex. This can be seen in the radar diagnoses of Tropical Storm Nuri by Raymond and López Carrillo (2011) (Fig. 4). The circulation calculated around a 3° latitude × 5° longitude box (~meso-α scale) has a midlevel maximum around 5 km. In the circulation budget analysis, the convergence term has maximum positive tendency between 4 and 5.5 km, with strong negative tendency above 6 km and weak negative tendency near the surface, which resembles the stratiform precipitation-dominant profile. In contrast, circulation around a 2° square box (~meso-β scale) has a maximum 2–3 km above the ground, and the convergence term contributes to positive tendency below the 6.5-km altitude and negative tendency above. The positive tendency is particularly strong below 3 km, indicating a deep convection-dominant profile. The different profiles of the different spatial averages, despite the tropical storm strength of the system, are consistent with our diagnoses of the numerical model simulation.

Vertical profiles of (first and third panels) circulation (planetary circulation in red and absolute circulation in blue) and (second and fourth panels) various circulation tendency terms for Nuri (2008) at the tropical storm stage derived from Eldora radar data. The left two panels show the averages over a 3° lat × 5° lon box, and the averages over a 2° square box are shown in the right two panels. [Adapted from Raymond and López Carrillo (2011).]

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Vertical profiles of (first and third panels) circulation (planetary circulation in red and absolute circulation in blue) and (second and fourth panels) various circulation tendency terms for Nuri (2008) at the tropical storm stage derived from Eldora radar data. The left two panels show the averages over a 3° lat × 5° lon box, and the averages over a 2° square box are shown in the right two panels. [Adapted from Raymond and López Carrillo (2011).]

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Vertical profiles of (first and third panels) circulation (planetary circulation in red and absolute circulation in blue) and (second and fourth panels) various circulation tendency terms for Nuri (2008) at the tropical storm stage derived from Eldora radar data. The left two panels show the averages over a 3° lat × 5° lon box, and the averages over a 2° square box are shown in the right two panels. [Adapted from Raymond and López Carrillo (2011).]

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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b. Thermodynamic structure of the pouch

The next question is why sustained deep convection is largely confined near the wave pouch center. The second hypothesis in Dunkerton et al. (2009) suggests that the wave pouch, as a region of approximately closed Lagrangian circulation, can retain moisture inside and prevent dry air intrusion to some extent. The wave pouch thus tends to have higher moisture content than its surrounding environment. However, it is not clear how thermodynamic conditions vary within the pouch. The dynamical aspect of the “sweet spot” has been explored by Wang et al. (2010a,b) and Montgomery et al. (2010b), but the thermodynamic conditions that help to make the pouch center a preferred location for genesis are not fully understood. To address this issue, the time–radius plots of saturation fraction (SF), the difference of equivalent potential temperature θ_e (Bolton 1980) between 950 and 700 hPa (hereafter θ_{e_diff}), and a parameter χ_m (to be defined below), are produced based on the model simulation.

SF is defined as the ratio of total precipitable water to saturated precipitable water from the surface to 300 hPa. It measures how close the column is to saturation. Raymond (2000) suggested that saturation deficit has a strong control on tropical precipitation. A significant correlation between precipitation rate and saturation fraction over the tropical oceans was confirmed by Bretherton et al. (2004) using Special Sensor Microwave Imager (SSM/I) satellite microwave radiometer data. As shown in Fig. 5a, SF is higher near the pouch center (the maximum SF is slightly off the center from 42 h onward, resembling an eye structure) and decreases with increasing radius. SF within the 100-km radius is generally above 90%, more than 10% larger than SF at radii larger than 300 km. In a nearly saturated environment, convective plume dilution from evaporation induced by entrained dry air is reduced, which is favorable for vigorous deep convection.

Time–radius plots of (a) saturation fraction (SF; %), (b) θ_e difference between 950 and 700 mb (950 minus 700 mb; K), and (c) χ_m in the numerical model simulation of Felix (see text for more details about this parameter).

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Time–radius plots of (a) saturation fraction (SF; %), (b) θ_e difference between 950 and 700 mb (950 minus 700 mb; K), and (c) χ_m in the numerical model simulation of Felix (see text for more details about this parameter).

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Time–radius plots of (a) saturation fraction (SF; %), (b) θ_e difference between 950 and 700 mb (950 minus 700 mb; K), and (c) χ_m in the numerical model simulation of Felix (see text for more details about this parameter).

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Figure 5b shows the difference in θ_e between the 950-hPa and the 700-hPa levels (i.e., θ_{e_diff}). The 700-hPa level is chosen to calculate θ_{e_diff} as it is the level of minimum θ_e in the model simulation (see Fig. 12 in Wang et al. 2010a). Note that θ_{e_diff} can be regarded as a measure of potential instability, and it can be easily derived from the dropsonde data to verify the model diagnosis. A smaller θ_{e_diff} implies potentially weaker downdrafts due to weaker evaporative cooling as well as weaker destructive impacts of downdrafts because of a weaker vertical gradient of θ_e (Rotunno and Emanuel 1987). As shown in Fig. 5b, within the 100-km radius from the pouch center, θ_{e_diff} is generally less than 7 K except in a small region over the pouch center after 39 h, and θ_{e_diff} increases with increasing radius. Further calculation (not shown) shows that the potential temperature difference between 950 and 700 hPa varies within 2 K within the 500-km radius. This suggests that the difference in θ_{e_diff} between the small and large radii (up to 10 K) is mainly due to the moisture distribution. The small θ_{e_diff} near the pouch center likely results from persistent convection, which moistens the middle troposphere, elevates the midlevel θ_e, and reduces the downdraft convective available potential energy (DCAPE) (Tory and Montgomery 2008; Tory and Frank 2010). It in turn provides a favorable environment for further convection. This is consistent with Smith and Montgomery’s (2012) diagnoses using dropsonde data from the PREDICT field experiment, which showed that the most prominent difference between the developing systems and one nondeveloping system is the smaller θ_e reduction between the surface and a height of 3 km in the developing systems. Davis and Ahijevych (2012) also found that a nondeveloping system has larger conditional instability than the developing disturbances they examined.

The parameter χ_m was introduced by Emanuel (1995) for the “Coupled Hurricane Intensity Prediction System” (CHIPS) model. It is defined as

View Expanded

where s_m and s_b are the moist entropies of the middle troposphere and the boundary layer, respectively, and s₀* is the saturation entropy of the sea surface. The moist entropy is defined as

View Expanded

where T is the temperature (K), C_p is the specific heat at constant pressure, L_υ is the latent heat of vaporization, q is the specific humidity, R_d is the gas constant for dry air, R_υ is the gas constant for water vapor, and r is the relative humidity.

Assuming that convective plumes have the same moist entropy as the boundary layer air, χ_m measures the ratio of the midlevel saturation deficit to the surface disequilibrium and indicates the relative importance of downdrafts and surface latent and sensible heat fluxes in controlling the subcloud layer entropy (Emanuel et al. 2008; Rappin et al. 2010). Using idealized model simulations, Rappin et al. (2010) showed that χ_m is closely related to the incubation time scale, or the time scale for a disturbance to moisten the middle troposphere and develop into an incipient tropical cyclone (Emanuel et al. 2008). As pointed out by Rappin et al. (2010), large values of χ_m are due either to large midlevel saturation deficit or to reduced surface disequilibrium (and thus weaker surface heat fluxes). In their idealized numerical simulations with relatively large values of χ_m, convective downdrafts reduce the column-integrated water vapor, and the associated low-level divergence weakens the surface vortex. As shown in Fig. 5c, χ_m is as low as 0.06 near the pouch center, in contrast to the larger values of 0.15–0.21 around the 300-km radius prior to genesis. This suggests the relatively weak detrimental impact of downdrafts near the pouch center.

In summary, the meso-β scale region near the pouch center is characterized by high saturation fraction, small θ_e difference between the surface and the middle troposphere, and small values of χ_m, which is suggestive of a short incubation time scale. These factors are all believed to be thermodynamically favorable for deep convection and tropical cyclone development. To examine how these thermodynamic conditions affect vertical motion, we constructed the contoured frequency by altitude diagrams (CFADs; Yuter and Houze 1995) of vertical velocity for radii less than 100 km and for the annulus between the 200- and 300-km radii, respectively (Fig. 6). The CFADs illustrate the frequency distribution of vertical velocity w of indicated values at different altitudes. The CFADs at t = 38 h (Fig. 6) show that vertical motion is strongly skewed toward updrafts, and updrafts stronger than 1 m s⁻¹ occur more frequently near the pouch center than within the annulus. It is interesting to note that strong downdrafts (>1 m s⁻¹) also occur more frequently near the pouch center than within the annulus. This suggests that the midlevel moistening and elevation of the midlevel θ_e do not necessarily suppress downdrafts (Nolan 2007). Instead, they promote vigorous updrafts by reducing convective plume dilution from evaporation due to dry air entrainment (James and Markowski 2010; Smith and Montgomery 2012).

Contoured frequency by altitude diagrams (CFADs) of vertical velocity at 38 h for the areas with (a) radii less than 100 km and (b) radii between 200 and 300 km in the numerical model simulation of Felix. To better display the large range of the frequency values, 10⁻⁴ is added to the frequency and then the base-10 logarithm is taken.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Contoured frequency by altitude diagrams (CFADs) of vertical velocity at 38 h for the areas with (a) radii less than 100 km and (b) radii between 200 and 300 km in the numerical model simulation of Felix. To better display the large range of the frequency values, 10⁻⁴ is added to the frequency and then the base-10 logarithm is taken.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Contoured frequency by altitude diagrams (CFADs) of vertical velocity at 38 h for the areas with (a) radii less than 100 km and (b) radii between 200 and 300 km in the numerical model simulation of Felix. To better display the large range of the frequency values, 10⁻⁴ is added to the frequency and then the base-10 logarithm is taken.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Another interesting feature in Fig. 6a is that intense updrafts near the pouch center peak at two altitudes, one at 3–4 km and the other at 11–12 km. The upper-level peak is also evident in the outer pouch region (Fig. 6b), but the peak in the lower troposphere is pronounced only near the pouch center. The updrafts peaking in the lower troposphere imply a strong low-level convergence, which is critical for spinup of the low-level circulation. Using idealized simulations, Raymond and Sessions (2007) suggested that lower-tropospheric cooling and upper-tropospheric warming reduce the altitude of the maximum vertical mass transport and enhances low-level convergence by stabilizing the air column. This implies that a midlevel vortex, presumably enhanced by stratiform precipitation, may play an important role in preconditioning the environment for tropical cyclone development (Raymond et al. 2011). To examine whether this is the case in the high-resolution model simulation, the change of virtual temperature from t = 22 h—that is, ΔT = T_υ(t) − T_υ(22 h)—is averaged within different radii. When averaged within the 50-km and the 100-km radii (Figs. 7a,b), the column is characterized by warming throughout the troposphere, with maximum warming up to 2 K around 700 hPa. Pronounced warming also occurs near the surface when averaged over the 100-km radii. When averaged within the 300-km radius, there is weak cooling (less than 0.3 K) around 800 hPa up to about 35 h and warming of much stronger magnitude in the boundary layer and in the middle and upper troposphere (between 400 and 700 hPa). The cooling around 800 hPa is likely due to stratiform processes, which make a relatively larger contribution to the total diabatic heating at the meso-α scale, as discussed before.

Time–height cross section of the virtual temperature difference (K) with respect to t = 22 h in the high-resolution simulation of Felix. The virtual temperature difference is averaged within different radii with respect to the pouch center: (a) 50, (b) 100, and (c) 300 km. Positive values are shaded.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Time–height cross section of the virtual temperature difference (K) with respect to t = 22 h in the high-resolution simulation of Felix. The virtual temperature difference is averaged within different radii with respect to the pouch center: (a) 50, (b) 100, and (c) 300 km. Positive values are shaded.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Time–height cross section of the virtual temperature difference (K) with respect to t = 22 h in the high-resolution simulation of Felix. The virtual temperature difference is averaged within different radii with respect to the pouch center: (a) 50, (b) 100, and (c) 300 km. Positive values are shaded.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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The midlevel warming revealed in Figs. 7a and 7b suggests that stability increases in the lower troposphere near the pouch center. It is plausible that updrafts would peak at a reduced altitude when air parcels become less buoyant in a stabilized atmosphere as suggested by Raymond and Sessions (2007). However, the change of static stability should be attributed to the midlevel warming instead of the low-level cooling. Besides, it is worth pointing out that Raymond and Sessions’ (2007) idealized simulations are based on variations of radiative–convective equilibrium (RCE) profiles. The RCE profile is an idealization of the quiescent or undisturbed tropical environment. In the current climate, most tropical cyclones originate from synoptic-scale disturbances, but only a small fraction of tropical synoptic-scale disturbances develop into tropical cyclones (Frank 1970). Raymond and Sessions’ comparison of a quiescent tropical environment with a disturbed environment suggests that the former is not favorable for tropical cyclogenesis, but they did not explain why some, but not all, disturbances provide a conducive environment to spawn a tropical cyclone or why there is a preferred location in tropical waves for TC genesis. This study and our other numerical modeling studies (Wang et al. 2010a,b; Montgomery et al. 2010b; Fritz and Wang 2012, manuscript submitted to J. Atmos. Sci.) suggest that the wave pouch center is the preferred location for genesis because 1) the pouch center serves a focal point for vorticity aggregation and convective organization and 2) the inner pouch region is protected from lateral dry air intrusion.

4. Balanced response to convective heating and stratiform heating

A fundamental aspect of tropical cyclone formation is the system-scale intensification, in which diabatic heating drives the transverse circulation and the associated low-level convergence intensifies the system-scale vortex by concentrating environmental vorticity. Convective precipitation and stratiform precipitation have different diabatic heating profiles. To better understand the spinup of the protovortex near the wave pouch center, we examine in this section the balanced responses to convective heating and stratiform heating using Eliassen’s theory for quasi-balanced vortices (Eliassen 1951). Eliassen’s model, which is often referred to as the Sawyer–Eliassen (SE) equation, has been used in many previous studies to examine the forced secondary circulation in a hurricane vortex (e.g., Willoughby 1979; Shapiro and Willoughby 1982; Schubert and Hack 1983). Willoughby (1979, 1990) showed that a balanced response is a good approximation if the tangential wind is much stronger than the radial wind. Montgomery et al. (2006) showed that the evolution of the system-scale vortex proceeds via approximate gradient wind and hydrostatic balance near the genesis time of a tropical depression in an idealized simulation. The balanced versus imbalanced responses for a mature storm were examined in detail by Bui et al. (2009). Here we will use the Sawyer–Eliassen equation to examine the transverse circulation associated with the wave pouch before the formation of a tropical depression.

We use the same form of the SE equation in height z coordinates as described in Bui et al. (2009):

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where ψ is the mass streamfunction, satisfying

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In Eqs. (5) and (6), r is the radius with respect to the pouch center; ρ is the density; χ is defined as the reciprocal of potential temperature (χ = 1/θ); υ and w are the tangential and vertical wind components, respectively; ζ = (1/r)∂(rυ)/∂r is the vertical component of relative vorticity; ξ = 2υ/r + f is twice the local absolute angular velocity; C = υ²/r + fυ is the sum of centrifugal and Coriolis force; Q is the diabatic heating rate; and F_λ represents the tangential components of frictional stress and the eddy momentum flux.

The WRF Model output is converted into height coordinates, and the relevant variables are then derived and transformed into cylindrical polar coordinates. The diabatic heating rate is taken as the residual term of the thermodynamic equation. Convective and stratiform heating rates are calculated for the convective regions and stratiform regions based on the precipitation partitioning in Wang et al. (2010b). The sum of the eddy momentum flux and frictional effect is calculated as the residual term in the azimuthal mean tangential momentum equation:

View Expanded

where the overbar denotes an azimuthal mean, the first term on the right-hand side of the equation is the local change of azimuthal mean tangential wind in the wave comoving frame, and ζ_a is the absolute vorticity [ζ_a = (1/r)∂(rυ)/∂r + f]. The SE equation is solved in a domain of 1000 km (radius) × 15 km (height), but the heating rate is gradually set to zero for radii larger than 500 km using an exponential decay function:

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where Q₀ is the heating rate derived from the thermodynamic equation, r₀ = 500 km, and D = 50 km. The azimuthal-mean temperature and tangential wind fields do not satisfy the ellipticity conditions at some points. To ensure robustness of the calculation, the equation is first normalized to remove symmetric instability and restore ellipticity in the domain of calculation following Bui et al. (2009). The forcing terms and the primary circulation are averaged over 24–40 h to derive the SE secondary circulation.

Figure 8 compares the azimuthally averaged radial wind structure in the WRF Model simulation with that derived from the SE streamfunction. The balanced response has midlevel inflow around 7 km and strong outflow in the upper levels, which is broadly consistent with the model simulation. Large discrepancies are found in the boundary layer, especially at large radii (>200 km) where the radial flow is comparable to the tangential wind (Willoughby 1990). The boundary layer inflow is underestimated in the balanced response (2 m s⁻¹ in the balanced response versus 3.5 m s⁻¹ in the WRF simulation). It is less extensive in the radial direction (see the −0.5 m s⁻¹ contour) and extends to a higher altitude within the 260-km radius compared to the WRF Model simulation. Similar differences between the balanced and unbalanced responses were also found by Bui et al. (2009) for a mature tropical storm. Figure 8 suggests that the balanced assumption breaks down in the frictional boundary layer, especially at the large radii, where rotation is weak and friction plays a more important role in force balance. On account of the discrepancies, the SE equation will be used only to understand the qualitative roles of the convective heating and stratiform heating in spinning up the TC protovortex at the pregenesis stage, and we mainly focus on the inner pouch region.

(a) Azimuthally averaged radial flow with respect to the pouch center during 24–40 h from the WRF Model simulation of Felix. (b) The radial flow derived from the balanced response when the Sawyer–Eliassen equation is forced with the total forcing (heat and momentum sources) derived from the WRF Model simulation.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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(a) Azimuthally averaged radial flow with respect to the pouch center during 24–40 h from the WRF Model simulation of Felix. (b) The radial flow derived from the balanced response when the Sawyer–Eliassen equation is forced with the total forcing (heat and momentum sources) derived from the WRF Model simulation.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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(a) Azimuthally averaged radial flow with respect to the pouch center during 24–40 h from the WRF Model simulation of Felix. (b) The radial flow derived from the balanced response when the Sawyer–Eliassen equation is forced with the total forcing (heat and momentum sources) derived from the WRF Model simulation.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Figure 9 shows the azimuthally averaged convective heating rate derived from the WRF Model simulation (Fig. 9a), the SE streamfunction (Fig. 9b), and the radial flow (Fig. 9c) and vertical wind (Fig. 9d) derived from the SE streamfunction. Strong convective heating occurs within the 100-km radius, with the maximum near but a little bit off the pouch center, and it decreases with increasing radius. Also note that the convective heating rate peaks around 3.5–4.0 km at all radii. The strong vertical gradient of heating rate below the peaking altitude implies strong low-level inflow.

(a) Convective heating rate derived from the WRF Model simulation (averaged over 24–40 h) (K h⁻¹); (b) streamfunction response (10⁸ kg s⁻¹); (c) radial inflow derived from the streamfunction (shading and white contours; m s⁻¹) and tangential velocity tendency associated with the radial flow (black contours; m s⁻¹ h⁻¹); and (d) vertical velocity (cm s⁻¹).

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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(a) Convective heating rate derived from the WRF Model simulation (averaged over 24–40 h) (K h⁻¹); (b) streamfunction response (10⁸ kg s⁻¹); (c) radial inflow derived from the streamfunction (shading and white contours; m s⁻¹) and tangential velocity tendency associated with the radial flow (black contours; m s⁻¹ h⁻¹); and (d) vertical velocity (cm s⁻¹).

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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(a) Convective heating rate derived from the WRF Model simulation (averaged over 24–40 h) (K h⁻¹); (b) streamfunction response (10⁸ kg s⁻¹); (c) radial inflow derived from the streamfunction (shading and white contours; m s⁻¹) and tangential velocity tendency associated with the radial flow (black contours; m s⁻¹ h⁻¹); and (d) vertical velocity (cm s⁻¹).

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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The balanced streamfunction response to convective heating (Fig. 9b) shows a strong cell centered near the 220-km radius and 4-km altitude. It is associated with inflow below 4 km and outflow above (Fig. 9c). The maximum inflow occurs below 2 km above the ground between the 50- and 250-km radii, with strong acceleration around the 250-km radius and deceleration around the 50-km radius. Recall that in cylindrical coordinates divergence can be written as

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where λ is the azimuthal. The contribution to the azimuthally averaged divergence from the last term is zero. The deceleration of the radial inflow contributes to strong convergence near the pouch center, while the acceleration of the radial inflow at the radii between 250 and 350 km leads to weak divergence due to the cancelation between the first and the second term in Eq. (9). Also shown in Fig. 9c is the contribution of the tangential acceleration by the radial flow . This momentum tendency term depends on both the radial wind and the absolute vorticity. With stronger cyclonic vorticity near the pouch center, the maximum tendency occurs around the 30-km radius, within the radius of the maximum tangential wind (~150 km), and decreases quickly with increasing radius. The spinup energy thus becomes more and more focused toward the heat source (Tory and Montgomery 2008), leading to the scale contraction from a meso-α wave pouch to an intense meso-β-scale protovortex.

Similar to the heating rate, the strong vertical motion is largely confined within the 50-km radius, in contrast to the relatively extensive low-level inflow. Wang et al. (2010a) showed that the immediate effects of tilting due to cloud updrafts are more local compared to the advective vorticity flux term in the vorticity budget equation. Figure 9d also shows weak downward motion at radii larger than 210 km, which tends to suppress convection.

The azimuthally averaged stratiform heating rate and the corresponding balanced response are shown in Fig. 10. The stratiform heating profile (Fig. 10a) is characterized by upper-level condensational heating and low-level evaporative cooling (the weak heating near the surface around the pouch center is likely due to the surface heat fluxes). Compared to the convective heating rate, the stratiform heating rate has a weaker magnitude and smaller radial and vertical gradients. The evaporative cooling is only about 20% of the convective heating in the lower troposphere.

As in Fig. 9, but for stratiform heating.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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As in Fig. 9, but for stratiform heating.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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As in Fig. 9, but for stratiform heating.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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The balanced streamfunction response to stratiform heating is characterized by a positive cell in the upper troposphere and a much weaker negative cell in the lower troposphere, both centered around the 200-km radius (Fig. 10b). Correspondingly, the transverse circulation features inflow between 4 and 8 km and outflow above (Fig. 10c). Below 4 km, weak outflow is found only near the pouch center between 1- and 4-km altitude and around the 200-km radius below the 2-km altitude. The vertical motion derived from the SE equation is characterized by moderate updrafts (~5 cm s⁻¹) above 5 km within the 200-km radius (Fig. 10d). Consistent with the weak negative streamfunction cell in the lower troposphere, downdrafts are quite weak (less than 1 cm s⁻¹) and are only present below 4 km between the 100- and 200-km radii. The midlevel inflow contributes to a positive tendency of tangential wind while the weak outflow does not induce any appreciable tangential wind tendency in the lower troposphere (Fig. 10c).

The precipitation partitioning in Wang et al. (2010b) follows the empirical algorithm of Tao et al. (1993). As noted by Wang et al. (2010b), the stratiform precipitation may be overestimated in the precipitation partitioning. Inclusion of convective heating in the stratiform category may offset the lower tropospheric cooling and enhance the upper tropospheric warming. To check the robustness of the results, we tested different criteria in the precipitation partitioning: grid points with rainfall rate at least twice as large as the average of their nearest eight neighbors are classified as convective cores, and columns with maximum upward vertical motions greater than 3 m s⁻¹ are also designated as convective, while in Wang et al. (2010b) a four-neighbor average and a 5 m s⁻¹ vertical motion threshold were used. With these changes, more grid points are designated as convective. The maximum stratiform heating in the upper troposphere is reduced to 0.6 K h⁻¹ while the stratiform cooling in the lower troposphere is also slightly reduced (not shown). The midlevel inflow in the balanced response to stratiform heating is weakened by 30%, and the low-level outflow remains weak but becomes more extensive, covering the 100–300-km radii below 2 km. This suggests that Fig. 10 provides a close estimate of the balanced response to the stratiform heating rate. Since the low-level outflow forced by the stratiform heating is much weaker than the midlevel inflow, it suggests that the stratiform heating contributes to the midlevel spinup without significantly spinning down the low-level circulation.

5. Thermodynamic conditions of the wave pouch revealed by PREDICT dropsonde data

In this section, the PREDICT dropsonde data are used to verify some modeling results. We will focus on pre-Karl, pre-Matthew, and ex-Gaston. The pre-Karl disturbance originated near the intertropical convergence zone (ITCZ) near northern South America. Despite its non-African origin, the disturbance resembled typical tropical easterly waves, propagating westward with an inverted-V pattern. It was surveyed by the PREDICT team for 5 consecutive days from 4 days prior to genesis (day −4) to the genesis day (day 0), with a double crew mission on 10 September (day −4). Matthew originated from an African easterly wave that left the west coast of Africa on 11 September [see the National Hurricane Center (NHC) tropical cyclone report on Matthew]. The wave first spawned Tropical Storm Julia near the Cape Verde Islands on 12 September. The ECMWF Interim Re-Analysis (ERA-Interim) data suggest that a weak remnant wave continued to propagate westward after Julia got detached from the wave on 16 September. The wave seemed to get enhanced through interaction with the ITCZ, and the pouch later spawned Matthew over the central Caribbean Sea. This sequence is consistent with one of the wave/vortex hybrid evolution scenarios summarized in Dunkerton et al. (2009), and it fits the type-B postgenesis wave evolution in Tyner and Aiyyer (2012). Matthew was surveyed for 4 days from 2 days prior to genesis (day −2) to 1 day after genesis (day +1). Gaston was a short-lived tropical storm. It was declared as a tropical depression on 1 September by the NHC but degenerated to a remnant low on the next day (see the NHC tropical cyclone report on Gaston). The disturbance continued propagating westward. PREDICT surveyed Gaston at the tropical cyclone stage on 2 September, and 1 day (day 1), 3 days (day 3), 4 days (day 4), and 5 days (day 5) after it weakened to a remnant low. Although possessing a well-defined wave pouch initially, ex-Gaston failed to reintensify because of persistent dry air intrusion above 500 hPa and vertical misalignment of the wave pouch (Rutherford and Montgomery 2011; Davis and Ahijevych 2012; Fritz and Wang 2012, manuscript submitted to J. Atmos. Sci.). It thus can be regarded as a nondeveloping system.

The propagation speed of a wave is estimated based on the Hovmöller diagram of the meridional wind from the ERA-Interim (Simmons et al. 2007). A pouch center is determined by the intersection of the wave critical surface and the trough axis. A 6-hourly pouch track is first produced, and an hourly pouch track is then derived using linear interpolation. To examine the thermodynamic structure of a wave pouch, the dropsondes for the three disturbances were categorized into two groups based on their proximity to the pouch center of the correspondent wave: within a 3° square box, and outside of the 3° box but within a 6° square box. The distribution of the dropsondes is shown in Fig. 11. Although a 3° square box slightly exceeds the meso-β spatial scale (20–200 km), it was chosen to ensure that enough drops are included in the first group. The calculations based on a 2° square box were found to be qualitatively similar to those based on a 3° box. The diagnosis based on the 3° box is presented below and is referred to as the inner pouch region (the meso-β scale), and the diagnosis outside of the 3° box but within a 6° box represents the outer pouch region, which is equivalent to the meso-α-scale area but excluding the inner pouch region.

Dropsondes for Gaston, Karl, and Matthew (a) within a 3° square box with respect to the pouch center (the inner pouch region) and (b) outside the 3° box but within a 6° box (the outer pouch region). The curves represent the pouch tracks. The number of the dropsondes for each storm is indicated in the parentheses. Note that in some flights (e.g., the last flight of Matthew) two dropsondes were released at the same location at two different times.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Dropsondes for Gaston, Karl, and Matthew (a) within a 3° square box with respect to the pouch center (the inner pouch region) and (b) outside the 3° box but within a 6° box (the outer pouch region). The curves represent the pouch tracks. The number of the dropsondes for each storm is indicated in the parentheses. Note that in some flights (e.g., the last flight of Matthew) two dropsondes were released at the same location at two different times.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Dropsondes for Gaston, Karl, and Matthew (a) within a 3° square box with respect to the pouch center (the inner pouch region) and (b) outside the 3° box but within a 6° box (the outer pouch region). The curves represent the pouch tracks. The number of the dropsondes for each storm is indicated in the parentheses. Note that in some flights (e.g., the last flight of Matthew) two dropsondes were released at the same location at two different times.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Figure 12a shows the dropsonde θ_e profiles for Karl on day −3, day −2, and day −1 averaged at the outer and inner pouch regions. Evident changes occur at the inner pouch region. From day −3 to day −2, θ_e increases by 1–3 K between 500 and 850 hPa and decreases by 1–3 K near the surface. From day −2 to day −1, θ_e increases above 950 hPa, with the maximum increase of 6 K around 850 hPa. Because of the monotonic increase of θ_e in the middle troposphere, the θ_e difference between the surface and 700 hPa decreased by 5–6 K from day −3 to day −1. At the outer pouch region, the change of θ_e is weaker and less systematic: θ_e increases 2–4 K below 850 hPa but decreases by 1–2 K between 500 and 600 hPa from day −3 to day −2; from day −2 to day −1, θ_e increases by 1–3 K between 520 and 850 hPa, without any substantial change below 850 hPa. Overall, the θ_e difference between the surface and 700 hPa increases at the outer pouch region from day −3 to day −1.

Vertical profiles of θ_e for Karl, Matthew, and Gaston. Solid lines represent the averages over the inner pouch region, and dashed lines represent the averages over the outer pouch region. Different colors represent different days.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Vertical profiles of θ_e for Karl, Matthew, and Gaston. Solid lines represent the averages over the inner pouch region, and dashed lines represent the averages over the outer pouch region. Different colors represent different days.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Vertical profiles of θ_e for Karl, Matthew, and Gaston. Solid lines represent the averages over the inner pouch region, and dashed lines represent the averages over the outer pouch region. Different colors represent different days.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Figure 12b shows the θ_e profiles for Matthew on day −2, day −1, and day +1. At the inner pouch region, there is up to a 5-K increase of θ_e from day −2 to day −1 above 800 hPa while θ_e remains visually the same at the surface, which indicates a 5-K reduction of the θ_e difference between the surface and the middle troposphere. From day −1 to day +1, θ_e increases significantly (up to 8 K) from 400 hPa all the way to the surface. At the outer pouch region, there is no systematic change of θ_e from day −2 to day −1, but θ_e increases significantly above 900 hPa from day −1 to day +1. Smith and Montgomery (2012) examined the system mean equivalent potential temperature of Matthew based on the PREDICT dropsonde data and found little change in mean θ_e from day −3 to day −1. This again suggests that spatial average over a large area may mask out the critical evolution of the protovortex near the pouch center.

Figure 12c shows the θ_e profiles for ex-Gaston 3 days (day 3), 4 days (day 4), and 5 days (day 5) after the storm weakened to a remnant low. Compared to Karl and Matthew, θ_e minimum occurs at a higher level in Gaston (450–500 vs 650–700 hPa). At the outer pouch region there is no systematic change in θ_e from day 3 to day 5. At the inner pouch region, θ_e at days 4 and 5 is reduced by 3–6 K above 500 hPa compared to day 3. It is also worth noting that θ_e at the outer pouch region is up to 15 K lower than that at the inner pouch region between 700 and 900 hPa on day 5. Further diagnosis indicates that the difference is mainly due to relative low moisture content at the pouch periphery induced by lateral dry air entrainment (not shown).

The two developing waves examined here, Karl and Matthew, are both characterized by the increase of the midlevel θ_e and decrease of θ_{e_diff} prior to genesis near the pouch center, with much weaker or little change away from the pouch center. In contrast, θ_e decreases between 400 and 700 hPa in Gaston near the pouch center. Figure 12 also reveals the difference in θ_e between the inner pouch region and the outer pouch region. In Karl, the midlevel θ_e at the inner pouch region becomes 2–3 K warmer than that at the outer pouch region 2 days prior to genesis, and the difference increases at day −1. In Matthew, there was no systematic difference in θ_e between the two spatial averages at day −2, but at day −1 the midlevel θ_e is up to 5 K warmer at the inner pouch region than that at the outer pouch region (the θ_e profiles at the two different spatial scales were brought closer to each other 1 day after the formation of Matthew). This is consistent with the diagnosis of the numerical model simulation in section 3 (Fig. 5b), and it suggests that 1) the thermodynamic conditions near the pouch center may be different from the pouch average, and 2) the thermodynamic conditions near the pouch center are critical for tropical cyclone development.

To examine what causes the change of the midlevel θ_e near the pouch center, the changes of specific humidity q, air temperature T, and θ_e for Karl from day −3 to day −1 at the inner pouch region (i.e., 3° × 3° box average) are shown in Fig. 13a. The value of θ_e increases significantly between 450 and 950 hPa, with the largest increase of 4–6 K between 750 and 900 hPa. The air temperature, however, decreases about 1 K below 700 hPa from day −3 to day −1, suggesting that the increase of equivalent potential temperature is due to the increase of specific humidity (~2 g kg⁻¹) or midlevel moistening. This is consistent with Nolan (2007), who showed in idealized simulations that the inner core approaches saturation prior to tropical cyclone genesis. Figure 13b shows the changes of q, θ_e, and T associated with Gaston from day 3 to day 5. Note that θ_e decreases between 700 and 400 hPa and increases below 900 hPa (there is also modest increase between 700 and 800 hPa). The changes of θ_e are accompanied by changes in specific humidity of the same sign, but the changes in air temperature are generally less than 1 K. This suggests that the midlevel drying is likely the cause for the nondevelopment of Gaston.

Changes of air temperature T (green; K), specific humidity q (red; g kg⁻¹) and θ_e (black; K) for Karl (day −1 minus day −3) and Gaston (day 5 minus day 3). The thin vertical line is the zero line.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Changes of air temperature T (green; K), specific humidity q (red; g kg⁻¹) and θ_e (black; K) for Karl (day −1 minus day −3) and Gaston (day 5 minus day 3). The thin vertical line is the zero line.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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Changes of air temperature T (green; K), specific humidity q (red; g kg⁻¹) and θ_e (black; K) for Karl (day −1 minus day −3) and Gaston (day 5 minus day 3). The thin vertical line is the zero line.

Citation: Journal of the Atmospheric Sciences 69, 8; 10.1175/JAS-D-11-0298.1

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6. Conclusions and discussion

The thermodynamic structure of the wave pouch and the spinup of the protovortex near the pouch center are examined in this study. The high-resolution simulation of pre-Hurricane Felix (2007) shows that the center of the wave pouch is characterized by high saturation fraction, small θ_e difference between the surface and the middle troposphere, and a short incubation time scale. Although stratiform precipitation prevails within the wave pouch prior to TC genesis (Wang et al. 2010b), these thermodynamic conditions help to make the pouch center particularly favorable for sustained deep convection. The associated low-level convergence can spin up a meso-β-scale protovortex near the pouch center, while the meso-α (or the pouch-mean) low-level convergence is relatively weak and may not be sufficient to offset the frictional spindown effect at the wave pouch scale. This results in different vertical structures and evolutions of the cyclonic circulation at different spatial scales (meso-β vs meso-α). The diagnosis of dropsonde data from the PREDICT field experiment confirmed the unique thermodynamic conditions near the pouch center and suggests that the critical information about the protovortex evolution near the pouch center may be masked out if a spatial average is taken over a large area within the wave pouch.

The Sawyer–Eliassen equation was applied at the pregenesis stage to better understand the system-scale spinup driven by diabatic heating. The high-resolution model simulation shows that convective heating peaks around the 4-km altitude and concentrates near the pouch center. The large radial and vertical gradients of the diabatic heating rate drive the transverse circulation. In particular, the strong vertical gradient of the diabatic heating in the lower troposphere concentrates the radial inflow near the surface. The convergence of the low-level radial inflow effectively intensifies the tangential circulation near the pouch center and induces the scale contraction from the meso-α scale to the meso-β scale. The balanced response to the stratiform heating features modest inflow in the middle troposphere and very weak outflow in the lower troposphere, which suggests that the stratiform process contributes to the midlevel spinup without significantly spinning down the low-level circulation.

A controversial issue about tropical cyclone genesis is the top-down development versus the bottom-up development. This study shows that different vertical development routes may take place at different spatial scales. At the meso-α scale (or the pouch scale), the low-level spinup is preceded by the midlevel spinup because of the frictional spindown effect and the relatively weak mean low-level convergence. This, however, does not necessarily imply that the midlevel spinup is essential for the development of the surface vortex, as suggested by some previous studies (Nolan 2007; Raymond et al. 2011). When zooming in at the meso-β scale, the surface spinup of the protovortex actually occurs before the midlevel spinup of the meso-α-scale circulation. It is also worth pointing out that the surface spinup at both spatial scales primarily results from the low-level convergence associated with deep convection.

The dynamic aspects of the tropical cyclone formation near the pouch center have been examined by Wang et al. (2010a,b) and Montgomery et al. (2010b). The favorable thermodynamic conditions near the pouch center are closely related to the dynamic and kinematic features of the pouch center. The region near the pouch center is characterized by strong rotation and weak shear/strain deformation, as well as weak relative flow in the comoving frame (the pouch center is a stagnation point by definition). Moisture lofted by deep convection from the boundary layer can thus accumulate in this region, while it is spread out or distorted into filaments away from the pouch center because of the strong strain rate. The middle troposphere can thus be moistened repeatedly by deep convection near the pouch center, especially for a vertically stacked deep wave pouch (Wang et al. 2012b). As suggested by the CFAD diagrams of vertical velocity, deep convection is more vigorous in such an environment, presumably due to reduced dry air entrainment or reduced moist entropy export, but downdrafts are not necessarily suppressed. Strong downdrafts actually occur more frequently in the meso-β region near the pouch center, but the detrimental impacts of downdrafts may be weakened by the weaker vertical gradient of θ_e. The low-level convergence associated with strong updrafts accelerates vorticity aggregation and intensifies the cyclonic rotation near the pouch center via the vortex-tube stretching effect. The resultant strengthened circulation can more effectively retain moisture and promote further deep convection. The dynamic and thermodynamic conditions thus reinforce each other and both contribute to convective organization and tropical cyclone formation near the pouch center.

The diagnosis of the PREDICT dropsonde data reveals a significant increase of midlevel θ_e and reduction of θ_e difference between the surface and the middle troposphere due to midlevel moistening 1–2 days prior to genesis near the wave pouch center, whereas significant changes may not take place in the outer pouch region before genesis. The midlevel moistening is presumably due to the accumulative effect of deep convection near the pouch center, and it may indicate convective organization near the pouch center and the impending tropical cyclone genesis. The forecast implication of this thermodynamic change merits further study.