The statistics of convective processes and vertical vorticity from the tropical wave to tropical cyclone stage are examined in a high-resolution simulation of Tropical Cyclone Fay (2008). The intensity of vertical velocity follows approximately the truncated lognormal distribution in the model simulation, which is consistent with previous observational studies. The upward motion at the pregenesis stage is weaker compared to mature hurricanes or midlatitude thunderstorms. The relatively strong upward velocities occupying a small areal fraction make a substantial contribution to the upward mass and moisture fluxes and condensation.
It is also found that upward motion and downward motion both intensify with time, but the former is stronger than the latter, and the mean vertical motion and the mean vertical mass flux thus increase with time. By contrast, the maximum anticyclonic vorticity is comparable to the maximum cyclonic vorticity in magnitude. Both cyclonic vorticity and anticyclonic vorticity intensify with time, but the former covers a larger areal fraction in the lower and middle troposphere and becomes dominant throughout the troposphere after genesis.
Sensitivity tests with different model resolutions were carried out to test the robustness of the results. When the horizontal grid spacing is reduced, the size of updrafts decreases and the number of updrafts increases, but the areal fraction of updrafts, the mean vertical velocity, and the mean vertical mass flux are rather insensitive to the model resolution, especially in the lower troposphere and when the model resolution is 1 km or higher. This may explain why models with relatively coarse resolution can simulate tropical cyclogenesis reasonably well.
As one of the most severe storm systems on the planet, tropical cyclones are characterized by strong rotation in the lower to middle troposphere and organized deep convection. The latent heat release drives the transverse circulation and plays an essential role in the system-scale intensification (e.g., Tory and Frank 2010; Montgomery and Smith 2010). Understanding the structure, evolution, and horizontal distribution of vertical motion and vertical vorticity is thus important for a better understanding of the formation and intensification of tropical cyclones. In particular, statistics of vertical motion help to understand the role of updrafts of different intensities in transporting mass, moisture, and energy during tropical cyclone development.
The statistics of updrafts and cloud properties in intense tropical cyclones and ordinary tropical convective systems have been examined in previous studies using flight-level data or radar data (e.g., Gray 1965; LeMone and Zipser 1980; Zipser and LeMone 1980; Jorgensen et al. 1985; Jorgensen and LeMone 1989; Black et al. 1996; Eastin et al. 2005; Anderson et al. 2005). López (1977) examined the frequency distributions of height, size, and duration of clouds and radar echoes in different convective conditions, and showed that they all follow the truncated lognormal distribution. He suggested that this could be attributed to the cloud formation process that obeys the law of proportionate effects. LeMone and Zipser (1980) and Zipser and LeMone (1980) examined the statistics of vertical motion associated with cumulonimbus clouds using aircraft data from the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) field experiment. They defined an updraft (a downdraft) as vertical motion that is continuously positive (negative) for at least 500 m and exceeds 0.5 m s−1 in magnitude, and an updraft (a downdraft) core was defined as the upward (downward) velocity that is continuously greater than 1.0 m s−1 for at least 500 m (these definitions were used in many subsequent studies). LeMone and Zipser (1980) showed that the size, magnitude, and mass fluxes of drafts and cores approximately follow the lognormal distribution, and found that only 10% of the updraft cores have mean velocity larger than 5 m s−1 in the middle troposphere.
Using aircraft data, Jorgensen et al. (1985) examined the properties of vertical motion in mature tropical cyclones (maximum surface wind speed > 50 m s−1). The vertical motion in the eyewall and rainbands of a mature hurricane were found to have an intensity distribution similar to those documented by LeMone and Zipser (1980) during the GATE field experiment, but the updraft and downdraft cores can be 2–2.5 times larger, which leads to stronger upward and downward mass transport. It was found that updraft mass flux at 1.5 km and below is mainly accomplished by small and weak cores; the mass transport above 1.5 km is distributed over a wide range of diameters and vertical velocities, and updraft cores >6 km in diameter make up only 5% of the population but contribute >40% to the upward mass transport. Using airborne Doppler radar data, Black et al. (1996) examined the characteristics of vertical motion in the eyewall, rainband, stratiform, and “other” regions of hurricanes. They showed that in all of these regions 70% of the vertical velocities are within −2 to 2 m s−1. Even in the eyewall region, only 5% of the vertical motions exceed 5 m s−1. It was also found that updraft cores outnumber the downdraft cores at least by a factor of two in all of the regions and that updraft cores are stronger and larger than downdraft cores.
The aforementioned observational studies suggested that (i) the vertical motion in tropical cyclones is not as vigorous as that in typical midlatitude thunderstorms and (ii) the relatively strong or large updrafts that account for only a small fraction of the total population make a considerable contribution to vertical mass transport. A similar finding was reported by Braun (2006) in the numerical model simulation of Hurricane Bonnie. It was shown that updrafts stronger than 4 m s−1 occupy less than 5% of the eyewall area but produce about half of the condensation. These studies, however, all focus on hurricanes or intense tropical storms, and the statistics of convective processes during the tropical cyclone formation stage has not been investigated. Given the strong influences of the ambient vertical vorticity on the development, intensity, structure, and lifetime of deep convection (Rozoff et al. 2006; Wissmeier and Smith 2011; Kilroy and Smith 2013), the effects of a tropical wave on convection are presumably different from those of an intense hurricane vortex. Satellite imagery shows that deep convection is intermittent and strongly modulated by the diurnal cycle at the early stage of tropical cyclone formation but is well organized and less influenced by the diurnal cycle at the tropical cyclone stage (deep convection organized by a low-level circulation is one of the criteria that forecasters use to declare the formation of a tropical cyclone). Therefore, the statistics of vertical motion for a mature hurricane may not apply to a tropical depression or a precursor disturbance.
The objective of this study is to examine the statistics of convective processes and vertical vorticity as a tropical wave transforms into a tropical cyclone. It would be ideal if one could examine the characteristics of vertical motion and clouds through observational diagnosis. However, most research or reconnaissance flights were deployed after a storm had formed, and radar or flight data at the pregenesis stage are too scarce to construct any meaningful statistics. We will thus employ numerical modeling. Numerical simulations admittedly have uncertainties due to model resolution and model physics (e.g., Bryan and Morrison 2012; Rotunno et al. 2009; McFarquhar et al. 2006; Fovell and Su 2007). Nevertheless, numerical models are an important and widely used tool for research and operational forecasting, and it is necessary to examine how well they represent convective processes. The statistics of convective processes and vertical vorticity can help us better understand the mass, moisture, and energy transport in a simulated storm, and comparison with previous observational studies, despite different convective conditions, will also help to evaluate the model physics.
The rest of the paper is organized as follows. The model simulation is described in section 2. Section 3 provides a brief description of the storm evolution. The statistics of vertical motion, vertical mass flux, vertical moisture flux, and as relative vorticity are described in section 4. The sensitivity to the model resolution is examined in section 5, and a summary is presented in section 6.
2. Model description and configuration
The Weather Research and Forecasting model (WRF) (Skamarock et al. 2008) version 3.2.1 was used to simulate the formation of Tropical Storm Fay (2008). The model is fully compressible and nonhydrostatic. The initial and lateral boundary conditions were derived from the 6-hourly Interim European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-Interim) data, which has a horizontal resolution of about 0.7°. The model was initialized at 0000 UTC 13 August 2008, 2.5 days prior to the formation of the tropical depression according to the National Hurricane Center (NHC) best track, and was integrated for 3 days. The precursor disturbance propagates westward at the speed of 6–7 m s−1, and travels about 1600 km in the 3-day simulation, which makes a fixed domain of high resolution computationally expensive. Automatic moving grids have been commonly used in numerical simulations or forecasting of tropical cyclones. With a moving nest, the center of a vortex is usually determined by the minimum of 500-hPa geopotential height or by the minimum sea level pressure (the WRF default setup; Skamarock et al. 2008), and the inner grid(s) of high spatial resolution moves automatically with the vortex. However, such vortex-tracking algorithms are designed to follow a well-defined tropical cyclone and are not suitable for a precursor wave with a weak midlevel circulation and no closed surface circulation. To simulate the formation of a tropical cyclone within an easterly wave, it is optimal to move the high-resolution, innermost grid(s) with the wave pouch1 center, which is the preferred location for genesis (Dunkerton et al. 2009; Wang et al. 2010a,b; Montgomery et al. 2010; Wang 2012). Wang et al. (2012) demonstrated that tracking the vorticity centroid is similar to tracking the wave pouch center and is superior to tracking individual vorticity maxima (the latter, like convective bursts, are more transient). This technique works well for disturbances propagating zonally and/or meridionally, and is useful not only for observational analysis but also for numerical simulations of propagating disturbances that require multiple nested grids. A vortex-tracking algorithm based on the low-level vorticity centroid was thus implemented in the WRF model to simulate the formation of Tropical Cyclone Fay.
The vorticity centroid is defined as
where ζ is relative vorticity and the subscript i denotes grid points within a certain domain. In this study the vorticity centroid was evaluated within a 200-km radius of the pouch center at the model level closest to 700 hPa, where the precursor wave has its maximum amplitude. The model was configured with four grids of 27–9–3–1-km resolutions. The outer two grids were fixed relative to the Earth, and the inner two grids moved with the vorticity centroid. The sizes of the four grids (from coarse to fine resolutions) are (147 × 92), (370 × 199), (322 × 292), and (643 × 526), respectively. The Kain–Fritsch scheme (Kain 2004) was applied in the outermost grid to resolve cumulus convection, and convection in the three inner grids (9–3–1 km) was resolved explicitly at the grid scale. Other model physics include Yonsei University (YSU) planetary boundary layer (PBL) scheme (Hong et al. 2006), WRF single-moment 6-class microphysics (WSM6; Hong and Lim 2006), the Rapid Radiative Transfer Model (RRTM) longwave radiation scheme (Mlawer et al. 1997), and Dudhia’s (1989) shortwave radiation scheme. The model has 40 vertical levels in a terrain-following, hydrostatic pressure vertical coordinate (a mass vertical coordinate), with 12 levels below 700 hPa.
To examine the sensitivity of the simulated vertical motion to the model horizontal resolution, two additional simulations were carried out. In the coarser-resolution (CR) test, the grid with 1-km resolution was dropped, and the model was integrated with three grids with the resolution of 27, 9, and 3 km, respectively. In the higher-resolution (HR) test, a fifth grid with 250-m horizontal resolution, which is comparable to some large-eddy simulations, was nested in the fourth grid of 1-km resolution. The CR simulation was integrated for 3 days as the control run. Because of the high computational cost, the HR test was integrated only for 6 h, 0000–0600 UTC 15 August (48–54 h in the control run). The CR and HR simulations both employed the same model physics as the control run.
3. Storm evolution
The evolution of the pre-Fay disturbance in the control simulation—in particular, the impacts of dry air intrusion on tropical cyclone formation—was examined by Fritz and Wang (2013). A brief description of the storm evolution is provided here, and interested readers are referred to Fritz and Wang (2013) for more details. Tropical Storm Fay (2008) developed from a tropical easterly wave that departed the African coast on 6 August 2008 (Brown et al. 2010). The zonal propagation speed of the wave is about −7 m s−1 as estimated from the Hovmöller diagrams of the meridional wind and total precipitable water (not shown) in the model simulation. This speed was used to determine the wave critical latitude and the wave pouch center. The latter is a stagnation point in the wave’s comoving frame of reference (Dunkerton et al. 2009), and is determined by the intersection of the wave trough axis and the wave critical latitude. The pouch tracks at 5-, 3-, and 1.5-km altitude are shown in Fig. 1a. The wave initially has a zonal path along 15.5°N. The track shifts northward around 57°W owing to the change of the mean flow and the advection by an anticyclone to its northeast. The wave then follows a nearly zonal path around 18°N in the last 2 days of the simulation. The wave is subject to moderate vertical shear (the shear between 850 and 200 hPa is ~8 m s−1 from west–southwest) during the simulation time period. The vertical displacement of the pouch center between 3 and 1.5 km is less than 150 km most of the time and reduces to less than 40 km shortly before genesis. The vertical displacement between 3 and 5 km is generally larger but also decreases approaching genesis. Recent studies (Dunkerton et al. 2009; Rappin et al. 2010; Wang et al. 2012; Rappin and Nolan 2012; Davis and Ahijevych 2012) suggested that the decrease of vertical tilt or the vertical alignment of a wave pouch is a critical process for TC genesis. Also shown in the figure are the locations of Tropical Cyclone Fay from the NHC best track after genesis. The errors between the best-track storm locations and the simulated pouch center locations at the corresponding times are less than 1°.
The time series of the simulated minimum sea level pressure (SLP) is shown in Fig. 1b along with the storm intensity from the NHC best-track data. The SLP of the simulated storm captures the trend of the best-track SLP but is slightly lower. According to the NHC best-track data, a tropical depression formed at 1200 UTC 15 August 2008 and was upgraded to a tropical storm 6 h later. In the model simulation, a tropical depression forms at 0600 UTC 15 August, when a closed circulation develops at the surface (not shown), and a tropical storm forms at 1200 UTC 15 August, when the 10-m maximum wind speed exceeds 17 m s−1. Both formation times in the model simulation are 6 h earlier than those in the NHC best-track data. The favorable comparison of the simulated storm track and intensity with the NHC best-track data suggests that the model simulation captures the pregenesis evolution reasonably well. The hourly output from the 1-km-resolution model grid will be used to examine the statistics of vertical velocity and other variables in the following sections. Wang (2012) showed that this meso-β-scale region near the pouch center is the preferred location for TC genesis. We will thus focus on the area within a 100-km radius of the 3-km pouch center (i.e., the inner pouch region) in most analyses if not specified otherwise, and the inner pouch and outer pouch regions are compared in section 4d.
4. Statistics of vertical motion and vertical vorticity
a. Frequency distribution of vertical motion and vertical mass flux
Observational studies based on radar data or flight data (e.g., Jorgensen et al. 1985; Black et al. 1996) have shown that vertical motion in mature tropical storms follows the lognormal distribution. As the first step, the probability distribution function (PDF) of the simulated vertical motion is examined. To facilitate comparison with previous observational studies, only vertical velocities with magnitude greater than 0.5 m s−1 are considered here. Figure 2 shows the upward and downward velocities at 5-km altitude in the lognormal display for 30–54 h (i.e., 24 h leading up to genesis) within 100-km radius of the pouch center. Upward and downward velocities approximately follow a straight line except at the tails of the distribution curves. This suggests that the vertical velocities in the model simulation follow a truncated lognormal distribution, which is consistent with previous observational studies (e.g., Jorgensen et al. 1985; Black et al. 1996).
The cumulative contoured frequency by altitude diagram (CCFAD) of the vertical velocity is shown in Fig. 3 (shading). CCFAD illustrates the cumulative probability of vertical motion at different heights, or (given the uniform model grid spacing) the areal fraction occupied by vertical velocities of magnitude less than or equal to a specific value at different heights. The distribution is derived from the vertical motion in the inner pouch region from 30 to 54 h, and the CCFAD of upward velocities (w > 0) and downward velocities (w < 0) are examined separately.2 What Fig. 3 does not show is the fractional coverage of upward or downward motion with respect to the total area, which will be examined later in section 4d.
As shown in Fig. 3, most upward and downward velocities have a very weak magnitude: 90% of upward velocities (w > 0) are less than 0.6 m s−1 below 9 km; w > 2 m s−1 accounts for less than 5% of upward velocities at all the levels; below 7 km, less than 1% of upward velocities exceed 5 m s−1. The statistics are consistent with previous findings that oceanic convective systems in the tropics have weaker vertical motion compared to continental or midlatitude thunderstorms (e.g., Kelley et al. 2010; Xu and Zipser 2012). The weak vertical motion can be attributed to a few factors, including different thermodynamic conditions between land and ocean (e.g., Williams and Stanfill 2002), smaller oceanic convective cores and the resultant more effective dilution by entrainment (e.g., Lucas et al. 1994; Zipser 2003), and stronger low-level convergence over land due to the surface roughness (Van Den Heever and Cotton 2007).
Figure 3 also shows that the top 10% of upward velocities have two maxima—one around 12 km and one around 4 km—with a minimum around 6 km. The midlevel minimum of vertical velocity has been reported in some previous studies and may result from the enhanced static stability at the freezing level, water loading, or midlevel dry air (Zipser and LeMone 1980; Lucas et al. 1994; Igau et al. 1999; Johnson et al. 1999; Fierro et al. 2008; Jensen and DelGenio 2006). The upper-level maximum is stronger than the low-level one, especially for the 99th percentile of vertical velocity. The broadening of the CCFAD with height suggests an acceleration of upward motion in the upper troposphere, which is consistent with analysis of observational data (Heymsfield et al. 2010; Hildebrand et al. 1996; Hildebrand 1998; Jorgensen et al. 1997) and can be attributed to the release of the latent heat of fusion (e.g., Zipser 2003; Romps and Kuang 2010; Fierro et al. 2012).
Compared to upward motion, downward motion is much weaker: 95% (99%) of the downward velocities are less than or equal to 0.6 (1.5) m s−1 below 10-km altitude. It is interesting to note that strong downdrafts occur roughly at the same altitudes as strong updrafts (i.e., around 12 and 4 km). Black et al. (1994) proposed that strong downdrafts in a hurricane may be caused by moist symmetric instability triggered by precipitation loading. It is not clear whether such a mechanism occurs in a weak vortex.
Figure 3 also shows the cumulative percentage of upward (downward) mass fluxes associated with upward (downward) velocities of magnitude less than or equal to a specific value.3 The CCFADs of vertical mass fluxes have a similar profile as the CCFADs of the vertical motion: where strong updrafts (downdrafts) occur more frequently, they make a larger contribution to the upward (downward) mass flux. Consistent with previous observational studies, the relatively strong upward velocities and downward velocities, despite occupying only a small areal fraction, make a considerable contribution to the vertical mass fluxes. For example, w > 1.5 m s−1, which accounts for less than 5% of the total upward motions at most levels, contributes 40%–50% to the total upward mass flux within a 100-km radius, the top 1% of upward velocities contribute about 20% to the total upward mass flux, and the top 5% of downward velocities contribute more than 20% to the total downward mass fluxes.
The mean vertical velocity averaged within a 100-km radius over 30–54 h is shown in the right panels in Fig. 3 (red curve). The mean vertical velocity is positive at all levels. It increases sharply with height below 3 km. From 3 to 6 km, there is no significant change. Above 6 km, w increases again with height and reaches a maximum at 11 km. The mean vertical velocity maximum occurs slightly below the altitude of the maximum upward motion (see the left panel in Fig. 3) because of the offset by the maximum downward motion at 12 km.
The upward, downward, and net vertical mass fluxes averaged within 100 km during the same time period are also shown in the right panels of Fig. 3. The mean upward and downward mass fluxes are calculated based on the formulas below:
where N is the total number of grid points within a 100-km radius. Note that mean upward and downward mass fluxes defined this way are additive: their sum equals the mean net vertical mass flux (a different average is defined with respect to the individual areas of upward and downward motion in Fig. 8, which yields a typical value of updraft or downdraft strength). The upward mass flux has a primary maximum around 2.5 km and a second maximum around 11 km. The downward mass flux is weaker than the upward mass flux at all the levels, and it has a low-level maximum around 1 km. The net vertical mass flux has two discernible maxima: one around 3 km and the other around 10 km. Because of cancellation by the downward mass flux, the net vertical mass flux is only about one-third of the mean upward mass flux in the lower troposphere, and it is also much weaker than the mean downward mass flux in the lower troposphere. This shows that the downward motion and downward mass flux are not negligible even in the inner pouch region. Also note that the profile of the net vertical mass flux is not as top heavy as the mean vertical motion profile because density decreases approximately exponentially with height. The vertical profile of the net vertical mass flux implies a layer of inflow below 3 km, a second layer of weak inflow between 7 and 10 km, weak outflow between 3 and 6 km, and strong outflow above 10 km. This complicated vertical structure may reflect a mix of different precipitation types, such as deep convection, congestus convection, and stratiform precipitation. The inflow branch in the middle to upper troposphere has been reported in some previous numerical studies (Montgomery et al. 2006; Nolan 2007; Wang et al. 2010b). Its elevation above the melting layer suggests that it likely results from acceleration of updrafts due to ice microphysical processes (e.g., Zipser 2003).
b. Vertical moisture flux and net condensation
The CCFADs of vertical moisture (water vapor) fluxes are shown in Fig. 4, superimposed on the CCFADs of upward and downward velocities. Similar to the vertical mass fluxes, the upward and downward moisture fluxes are examined separately, and the CCFADs show the cumulative contribution to the total upward (downward) moisture flux by upward (downward) velocities of magnitude less than or equal to a specific value. The strong upward velocities of a small population make a significant contribution to the total upward moisture flux: the top 10% of upward velocities contribute more than 60% to the total upward moisture transport above 2.5 km and more than 50% below 2.5 km. The CCFAD of the downward moisture flux is much narrower than that of the upward moisture flux. Nevertheless, the top 5% of the downward velocities (stronger than ~0.5 m s−1) contribute 20%–30% to the total downward moisture flux.
The upward, downward, and net vertical moisture fluxes averaged within a 100-km radius are shown in the right panel of Fig. 4. The upward moisture flux is stronger than the downward moisture flux at all levels. Both upward and downward moisture fluxes peak around 1 km, with the former slightly above the latter, and decrease sharply with height above 1 km. The net vertical moisture flux peaks around 2.5 km and is positive throughout the troposphere. Because of the offset by the downward moisture flux, the net moisture flux is only about 20% of the upward moisture flux at 1-km altitude.
The cumulative contribution of vertical velocities of different intensities to the total condensation is shown in Fig. 5. The condensation field is directly output from WRF. Positive values mean that the rate of condensation and deposition exceeds the rate of evaporation and sublimation in a model grid cell, and are referred to as condensation for simplicity. Negative values indicate that the rate of evaporation and sublimation exceeds the rate of condensation and deposition, and are referred to as evaporation. The total condensation is the sum of positive values within a 100-km radius, excluding grid points with negative values. Although condensation is generally associated with upward motion, a small fraction of weak positive condensation occurs at grid points with downward motion. The CCFAD of condensation shows the contribution to the total condensation by vertical velocities less than or equal to a specific value (including downward velocities associated with positive condensation values).
The CCFADs of vertical motion and condensation show that strong updrafts, despite occupying a small fraction of area, make a significant contribution to the total condensation: the top 1% of upward motion contributes 20%–30% to the total condensation below 5-km altitude and more than 30% above 5 km; w > 1 m s−1 accounts for less than 10% of the upward motion population but contributes more than 70% to the total condensation.
The vertical profiles of the mean condensation (average of positive values only), the mean evaporation (average of negative values only), and the mean net condensation (average of both positive and negative values) within a 100-km radius are shown in the right panel. The mean evaporation is much weaker than the mean condensation except near the surface. The profile of the mean net condensation resembles that of the mean condensation and has a primary maximum around 4.5 km. The increase of the net condensation with height in the lower troposphere implies a positive potential vorticity tendency. Between 4.5 and 6 km and above 10 km, the net condensation decreases with height, contributing to a negative potential vorticity (PV) tendency.
c. Frequency distribution of relative vorticity
The frequency distributions of cyclonic vorticity and anticyclonic vorticity (shading) and their contribution to the total cyclonic and anticyclonic vorticity (contours) are shown in Fig. 6. Prior to genesis, cyclonic vorticity stronger than 1 × 10−3 s−1 makes up less than 10% of the population, but contributes 40%–50% to the total cyclonic vorticity; cyclonic vorticity stronger than 1.5 × 10−3 s−1 accounts for less than 5% of the population at most levels but contributes more than 30% to the total cyclonic vorticity within a 100-km radius. It is worth pointing out that these extreme vorticity anomalies are not necessarily related to deep convection or vortical hot towers (VHTs; Hendricks et al. 2004; Montgomery et al. 2006). As shown in Wang (2014), strong vorticity anomalies can also be induced by cumulus congestus.
As shown in section 4b, downdrafts tend to be much weaker than updrafts, and the CCFADS of vertical velocity are strongly asymmetric (Fig. 3), but this is not the case for relative vorticity. Figure 6 shows that strong anticyclonic vorticity is comparable to strong cyclonic vorticity in magnitude, but the occurrence of the former is less frequent than the latter as shown later in section 4d. The vertical profile of the anticyclonic vorticity distribution is similar to that of the cyclonic vorticity distribution: the 95th percentile increases from 1 × 10−3 s−1 at the surface to 1.8 × 10−3 s−1 at ~2.5-km altitude and then decreases slightly with altitude from 2.5 to 8 km. A secondary maximum occurs around 11 km. The similarity of the CCFADs of cyclonic and anticyclonic vorticity can be explained by the tilting term, which tends to produce a vorticity dipole pattern for a localized updraft (Montgomery et al. 2006; Tory and Frank 2010).
The cyclonic, anticyclonic, and net relative vorticity averaged within a 100-km radius are shown in the right panel of Fig. 6. Cyclonic vorticity and anticyclonic vorticity both have two maxima: one around 3 km and the other around 11 km. The mean cyclonic vorticity is stronger than the mean anticyclonic vorticity below 11 km and is weaker above. The mean net relative vorticity is thus cyclonic below 11 km and anticyclonic above, with a clear maximum below 1 km. As shown in Wang (2012), the vertical profile of circulation (or vorticity) depends on the spatial scale. The warm-core structure is most pronounced at the meso-β scale near the pouch center. At the meso-α scale, the vortex may remain a cold core structure in the lower troposphere prior to genesis.
d. Temporal evolution
Time–height cross sections of vertical motion, vertical mass flux, vertical moisture flux, and net condensation averaged within a 100-km radius are shown in Fig. 7. The mean vertical velocity is characterized by a top-heavy profile most of the time, which can be attributed to vigorous deep convection and stratiform precipitation processes (Wang et al. 2010b). Heymsfield et al. (2010) found that vertical velocity in tropical cyclone convection peaks near 12-km altitude. Wang (2014) shows that deep convection has two maxima: one in the upper troposphere (~11 km) and one in the lower troposphere (4–5 km). The presence of stratiform precipitation enhances the upper-level peak and weakens the low-level peak, contributing to the top-heavy profile (Wang et al. 2010b). The maximum mean vertical velocity increases from less than 0.05 m s−1 at 30 h to ~0.4 m s−1 at 60 h. The mean vertical motion undergoes high-frequency fluctuations with a period of 3–6 h. In each fluctuation cycle, vertical motion develops first in the lower troposphere and then extends upward. An upper-level maximum appears 2 or 3 h later as the low-level vertical motion is weakening, leading to a top-heavy vertical profile. This evolution is consistent with the natural life cycle of clouds, during which shallow convection evolves into deep convection and then transitions into stratiform precipitation with high anvil clouds. The high-frequency fluctuations are still detectable after genesis (i.e., after 54 h), but the vertical motion becomes stronger, more vertically extensive, and more persistent. The maximum w shifts to a lower altitude (~8 km) after 65 h, which is likely due to the land impacts as the wave pouch moves over Puerto Rico and Hispaniola (see the pouch track in Fig. 1a).
The mean vertical mass flux also has high-frequency fluctuations prior to genesis, and becomes much stronger after genesis (t = 54 h). It has two maxima: one around 3 km and the other around 10 km. The strong vertical gradient of the upward mass flux near the surface implies strong mass convergence within the boundary layer, which can effectively intensify the low-level circulation. The elevated maximum at 10 km implies another inflow layer in the middle to upper troposphere.
The vertical moisture flux (Fig. 7c) has a maximum around 2 km and decreases sharply upward. Similar to the vertical motion and the vertical mass flux, the moisture flux also undergoes high-frequency fluctuations. At the peak of the fluctuations, upward moisture flux extends to a higher altitude and moistens a deeper layer. The column moistening in turn creates a more favorable environment for deep convection by reducing lateral dry air entrainment (e.g., Redelsperger et al. 2002; Derbyshire et al. 2004). High humidity in the lower troposphere also helps to support deep convection via moisture entrainment (Holloway and Neelin 2009; Wang 2014).
The time–height evolution of net condensation is shown in Fig. 7d. The net condensation evolution has similar high-frequency fluctuations as vertical motion. It peaks at three approximate altitudes—2, 4, and 8 km—corresponding to the trimodal distribution of tropical convection (e.g., Johnson et al. 1999; Wang 2014) and the maximum around 4 km (the freezing level) is most pronounced. The net condensation is enhanced significantly above 4 km around t = 54 h. The vertical gradient of diabatic heating rate implies a strong positive tendency of potential vorticity below the freezing level and near the surface.
The time–height cross section of the areal fraction of upward motion (w > 0) within a 100-km radius is shown in Fig. 8a as a complement to the CCFADs shown in section 4a. Upward motion is dominant above 6 km, and downward motion occurs more frequently below 6 km. The particularly low (high) areal fraction of updraft in the lower (upper) troposphere during 15–24 h indicates the prevailing occurrence of stratiform precipitation and can be attributed to the impacts of a transient dry-air intrusion event (Fritz and Wang 2013). The areal fraction of upward motion is similar to Jorgensen et al.’s (1985) finding for mature tropical storms using aircraft data that updrafts cover a larger areal fraction than downdrafts above 5 km and a smaller areal fraction below 3 km.
The time series of the mean upward and downward velocities within a 100-km radius at 3-km altitude are shown in Fig. 8b. Different from the right panels in Figs. 3–6, here upward and downward velocities are normalized by the areas occupied by upward and downward motion, respectively. Both the mean upward and downward velocities increase with time and have strong fluctuations, but the magnitude of the mean upward velocity is much stronger than that of the mean downward velocity. The mean vertical velocity and vertical mass flux are thus upward and increase with time (Fig. 7).
The evolution of relative vorticity averaged within a 100-km radius is shown in Fig. 9a. The vorticity maximum occurs at 4–5 km prior to 26 h, which is consistent with the midlevel convergence implied by Fig. 8a. From 30 to 60 h, the maximum vorticity occurs below 2 km. The mean vorticity gradually increases with time in the lower troposphere from 26 to 48 h, followed by a rapid intensification of vorticity throughout the troposphere. The rapid intensification of vorticity occurs at the same time with the significant increase in the mean vertical mass flux and the mean net condensation shortly before genesis (Fig. 7). Recent studies have shown that the low-level convergence is the most effective way to intensify the surface circulation (e.g., Tory and Frank 2010; Wang et al. 2010a; Fang and Zhang 2010; Raymond and López Carrillo 2011). The potential vorticity budget analysis in Wang (2014) revealed that cumulus congestus makes the major contribution to spinning up the surface circulation prior to genesis while deep convection plays a dominant role in amplifying the circulation in the free troposphere.
The areal fraction of cyclonic vorticity within a 100-km radius is shown in Fig. 9b. Cyclonic vorticity covers a larger fractional area (>55% of the total area) than anticyclonic vorticity below 6 km all the time. The two compete in the upper troposphere, and cyclonic vorticity becomes dominant from the surface up to 14 km by 60 h, which is consistent with the upward development of the TC vortex. The mean cyclonic vorticity and the mean anticyclonic vorticity (Fig. 9c) increase with time at the same pace, and the magnitudes of the two are comparable, which is consistent with the CCFADs of relative vorticity (Fig. 6). The increase in the areal coverage of cyclonic vorticity can be attributed to the increasing background (or system-scale mean) cyclonic vorticity owing to the system-scale convergence (e.g., Wang et al. 2010a). Since the background (or mean) vorticity (Fig. 9a) is one order of magnitude smaller than the extreme cyclonic or anticyclonic vorticity anomalies, it does not change the symmetric appearance of the CFADs (Fig. 6), but makes the frequency distribution (or areal coverage) increasingly skewed toward cyclonic vorticity. Figures 9b and 9c suggest that the increase in the mean relative vorticity is mainly due to the system-scale convergence of cyclonic vorticity rather than intensification of extreme vorticity anomalies as strong vorticity dipoles generated by the tilting term cannot increase the system-scale intensity. Wang (2014) showed that cumulus congestus, with a bottom-heavy heating profile, can effectively drive the low-level convergence and enhance the PV production near the surface.
The CCFADs in the previous subsections were based on the 1-day time period prior to genesis. To examine better the temporal evolution of extreme vertical velocity and vorticity as the disturbance transitions from a tropical wave to a tropical storm, the probability distributions of w at 4 km and relative vorticity at 2 km in the inner pouch region are shown for 24–72 h in Fig. 10. Again, positive and negative values are examined separately for both variables.
A striking feature of Fig. 10a is that the cumulative distributions of upward velocity and downward velocity both broaden with time in the inner pouch region, suggesting that upward and downward velocities both intensify with time. This is consistent with the findings of some previous studies. In the simulation of Felix (2007), Wang et al. (2010a) showed that stratiform downward motion and convective upward motion both intensify approaching the genesis time. Nolan (2007) found the frequency of deeper and stronger updrafts increases with time while the frequency of downdrafts remains essentially unchanged in his idealized simulations. An evident and important feature in Figs. 10a and 3 is that the PDF of updrafts spreads to higher values than that of downdrafts. In other words, updrafts tend to be stronger than downdrafts. The mean vertical velocity and vertical mass flux thus increase with time (Fig. 7). Superimposed on the intensification tendency are strong fluctuations and a weak but discernible diurnal cycle peaking at 0900–1200 UTC (5–8 a.m. local time).
The cumulative distributions of cyclonic and anticyclonic vorticity also broaden with time, and the magnitude of extreme anticyclonic vorticity remains comparable to that of extreme cyclonic vorticity. It is interesting to note that the cumulative distributions of low-level cyclonic vorticity and anticyclonic vorticity do not show a diurnal cycle and that the fluctuations of ζ are not as strong as those of w. This is likely due to the different lifetimes of updrafts and vorticity anomalies. Although convective updrafts are short lived, the resultant vorticity anomalies can last much longer (Montgomery et al. 2006; Fang and Zhang 2011; Wissmeier and Smith 2011).
Also shown in Fig. 10 are the cumulative distributions of vertical velocity (Fig. 10c) and relative vorticity (Fig. 10d) between 150- and 250-km radii (i.e., the outer pouch region). In contrast to the inner pouch region, the vertical motion and relative vorticity in the outer pouch region are modulated strongly by the diurnal cycle and do not have an intensification trend, and upward motion and cyclonic vorticity are also much weaker in the outer pouch region. This supports the finding of Wang (2012) that the inner pouch region is particularly favorable for deep convection and tropical cyclogenesis. The different evolution and distribution of vertical motion and vertical vorticity between the inner and outer pouch regions can be attributed to the different thermodynamic conditions between the two regions (Wang 2012). As shown in Fig. 11a, equivalent potential temperature (θe) near the surface is lower in the inner pouch region than in the outer pouch region. The θe is modulated by the diurnal cycle, and areas of low θe propagate outward. At 600 hPa (Fig. 11b), although a diurnal cycle is discernible prior to 60 h, a more notable feature is the increase in θe in the inner pouch region, and θe is distinctively higher in the inner pouch region than in the outer pouch region. As a result, the θe difference between 950 and 600 hPa is 5–6 K smaller in the inner pouch region than in the outer pouch region (radii larger than 200 km), which is consistent with the dropsonde analysis in Wang (2012). The lower surface θe and the higher midlevel θe in the inner pouch region can be attributed to the strong vertical motion near the pouch center (Figs. 10a and 10c). While vigorous convection moistens the middle troposphere and increases the midlevel θe, convective downdrafts bring air of lower θe to the boundary layer and reduce the surface θe.
The θe difference between the surface and the middle troposphere is a measure of potential instability. The decrease in the θe difference seems to contradict the intensification of the vertical motion, but the effective cloud buoyancy is strongly modulated by dilution of cloud cells owing to entrainment (Zipser 2003; Romps and Kuang 2010; Molinari et al. 2012). A small θe, difference between the surface and 600 hPa due to midlevel moistening implies weaker dry-air entrainment. With weaker dry-air entrainment, updrafts have a better chance to extend above the freezing level and gain further buoyancy from the release of latent heat of fusion. It is worth noting that the diurnal cycle of θe difference is detectable in both the inner and outer pouch regions, but the diurnal cycle of vertical velocity is much weaker in the inner pouch region. This suggests that the θe difference may work as a threshold to modulate convection. If the θe difference is below a certain threshold or the midlevel humidity is above a certain threshold, lateral entrainment may not significantly reduce the parcel buoyancy, and vertical motion may become relatively insensitive to the environmental humidity. Further research is warranted to better understand the impacts of environmental humidity on cloud evolution.
5. Sensitivity tests
The sensitivity of a simulated convective system to the model resolution has been reported in some previous studies (e.g., Weisman et al. 1997; Bryan et al. 2003). Bryan and Morrison (2012) showed that the simulated squall lines are sensitive to both the model microphysics schemes and horizontal resolution. Rotunno et al. (2009) examined the sensitivity of the tropical cyclone intensity to the model resolution in idealized large-eddy simulations (LES). It was found that the storm intensity increases when the model grid interval decreases from 1.67 km to 185 m and starts to decrease when the grid interval is further reduced to 62 m. Previous observational analysis showed that the median diameter for updrafts and downdrafts is about 2 km in mature tropical cyclones (e.g., Jorgensen et al. 1985). To capture the realistic structure of individual convective cells, one will have to employ a large-eddy simulation, which is extremely computationally expensive for a real-storm simulation. Nevertheless, most models with horizontal resolution of O(1) km simulate or forecast the evolution of a tropical cyclone reasonably well. This implies that the realistic simulation of individual convective cells is not necessary for realistic simulation or skillful forecast of tropical cyclones. What really matters is the bulk effects or the statistics of convective updrafts.
In this section we will examine the sensitivity of vertical velocities to the model horizontal resolution. The vertical velocities from the fifth grid output (250-m resolution) in the HR run and the third grid output (3-km resolution) in the CR run are compared with those from the fourth grid output (1-km resolution) in the control run for 51 to 54 h below. The 3-km pouch track and the time series of SLP from the CR run are shown in Fig. 1, and both are similar to the control run (the pouch track and the storm intensity are not shown for the HR run owing to the short simulation time period). Following LeMone and Zipser (1980), an updraft is defined as w > 0.5 m s−1, but no size limit is applied. A continuous area of w > 0.5 m s−1 at pressure surfaces is defined as one updraft, and its equivalent diameter is defined as , where A is the area of the updraft. The minimum area of an updraft in the control run is 1 km2 (one grid cell), and the minimum equivalent diameter that can be represented in the control run is = 1.13 km. Similarly, the minimum equivalent diameter of an updraft is = 0.28 km in the HR run and 3.38 km in the CR run. Figures 12a and 12b show the mean diameter and the total number of updrafts at different altitudes within a 100-km radius from the three simulations. In the control run (black curves), the mean diameter of updrafts varies between 2.5 km (near the surface) and 4.8 km (at 12 km). Similar to the mean vertical motion, the mean diameter reaches its maximum in the upper troposphere (12 km). The increase of the mean diameter with height between 7 and 12 km suggests a tendency for updrafts to organize into larger entities in the middle to upper troposphere, which is consistent with Jorgensen et al. (1985). The number of updrafts has a primary maximum near 1-km altitude and a secondary maximum around 11 km. The updraft number drops from ~210 at 1 km to ~100 at 6 km. From 9 to 11 km, both the mean diameter and the number of updrafts increase with height, which is consistent with the peaking of the mean vertical motion at 11 km.
Compared to the control run, the mean diameter of updrafts is increased by a factor of 3 in the CR run (blue curves) and is reduced by a factor of 3 in the HR run4 (red curves), but the vertical profiles have a similar shape. The number of updrafts is reduced by a factor of 6 in the CR run and increased by a factor of 5 in the HR run. The vertical profiles are again similar to that in the control run. What is probably more important is that the areal fraction of updrafts (or upward mass flux) in the CR and HR runs is close to that in the control run (Fig. 12c). The areal fraction of updrafts is only slightly larger (smaller) in the HR run than in the control run below (above) 9 km, and is slightly smaller (larger) in the CR run below (above) 5 km.
The CCFADs of upward and downward velocities for the three simulations are compared in Fig. 13a. The CCFADs in the HR simulation are close to those in the control run, except that the strong upward and downward motions occur slightly more frequently in the lower troposphere in the HR run while the 99th percentile upward velocity is weaker in the HR run above 4 km. The mean upward motion in the HR run is slightly but consistently stronger than that in the control run below 8 km and weaker above 8 km (Fig. 13b). The mean downward motion in the HR run is slightly stronger than or visually equal to that in the control run. Interestingly, the mean vertical motion in the HR simulation is nearly identical to that in the control below 6 km but slightly weaker above 6 km. This is probably due to the larger size of the updrafts in the control run, which reduces the effects of lateral dry air entrainment and enables updrafts to penetrate through a higher altitude.
Much larger differences are found between the CR run and the control run. Compared to the control run, the CCFADs of upward and downward velocities show that the frequency of occurrence of strong updrafts is overpredicted (underpredicted) above (below) 2.5 km and that frequency of strong downdrafts is underpredicted in most levels in the CR run. For example, the 95% upward motion at 11 km is < 2.5 m s−1 in the control run but is close to 4 m s−1 in the CR run. Consequently, the mean upward motion is stronger (weaker) above (below) 2.5 km in the CR run than in the control run, and the mean downward motion is slightly weaker in the CR run at all levels (Fig. 13d). The mean vertical motion is weaker in the CR run than in the control run below 4 km and significantly stronger in the middle and upper troposphere (the maximum mean vertical motion in the CR run is twice as strong as that in the HR run).
The comparison between the three experiments suggests that 3-km resolution is too coarse to resolve the statistics of vertical motion but the simulation starts to converge at 1-km resolution. The statistics derived from the control run in section 4 is thus believed to be robust with respect to the model resolution. It is worth noting that the mean vertical motion and vertical mass flux (the latter is not shown) are relatively insensitive to the model resolution, especially in the lower troposphere. This explains why models with relatively coarse resolution can capture the tropical storm evolution reasonably well (see Fig. 1).
Statistics of convective processes and vertical vorticity from the tropical wave to tropical cyclone stage are examined in a high-resolution simulation of Tropical Cyclone Fay (2008). To resolve convection better near the pouch center, a vortex-tracking algorithm based on the vorticity centroid was implemented in WRF, and the two inner model grids (with resolutions of 3 and 1 km, respectively) move automatically with the pouch center. This model configuration allows reasonable representation of both the synoptic-scale wave and the convective processes near the wave pouch center. The model simulation spans from the tropical wave stage (more than 2 days prior to the genesis) to the tropical cyclone stage and captures the formation of a tropical depression vortex within the synoptic-scale wave. Two sensitivity runs were carried out to examine the robustness of the results. In one experiment, the fourth grid was dropped and the model was run with three grids (with the resolution 27, 9, and 3 km, respectively). In the other experiment, a fifth grid with 250-m resolution was added.
The major findings are summarized below:
The intensity of upward and downward velocities approximately follows the truncated lognormal distribution in the model simulation, which is consistent with previous observational studies.
The upward motion at the pregenesis stage is relatively weak compared to mature hurricanes or midlatitude thunderstorms. The vertical velocity w > 1 m s−1 accounts for less than 10% of the total upward motion population at all the vertical levels, but makes a substantial contribution to the upward mass flux (40%–50%), the upward moisture flux (50%–60%), and condensation (~70%) in the inner pouch region.
Upward motion tends to occur more frequently than downward motion above 6 km and slightly less frequently below 6 km. Upward motion and downward motion both intensify with time, but the former is stronger than the latter. The mean vertical velocity and the mean vertical mass flux are both upward and increase with time.
The top 10% of cyclonic vorticity contributes 40%–50% to the total cyclonic vorticity in the inner pouch region. The maximum anticyclonic vorticity is comparable to the maximum cyclonic vorticity in magnitude, and both intensify with time, but cyclonic vorticity covers a larger areal fraction below 6 km. After the disturbance intensifies to a tropical cyclone, cyclonic vorticity occurs more frequently than anticyclonic vorticity throughout the troposphere.
The size of updrafts decreases and the number of updrafts increases with decreasing model grid spacing, but the areal fraction of updrafts (w > 0.5 m s−1), the mean vertical velocity, and the mean vertical mass flux are relatively insensitive to the model resolution, especially in the lower troposphere and when the model resolution is 1 km or higher.
The different statistics and evolution of vertical velocity and vertical vorticity between the inner and outer pouch regions support the finding of Wang (2012) that the thermodynamic conditions in the inner pouch region are particularly favorable for deep convection and TC genesis.
The probability distribution of vertical velocity was examined at different heights using cumulative CFADs, with or without weighting factors representing mass and moisture fluxes, condensation processes, and so on. However, the vertical profile of individual updrafts was not examined. An updraft with weak magnitude in the lower troposphere may obtain a large intensity at the upper troposphere owing to the latent heat of fusion. In the CCFADs it will fall into the weak vertical velocity regime in the lower troposphere and into the strong vertical velocity regime in the upper troposphere. The CCFADs do not provide information on the vertical extent of updrafts either. The role of different types of convection in tropical cyclone formation is examined in a companion study (Wang 2014).
This research was supported by the National Science Foundation Grants AGS-1016095 and AGS-1118429. The author would like to thank Dr. Tim Dunkerton and three anonymous reviewers for the helpful comments on an early version of the manuscript and Isaac Hankes for proofreading the manuscript. Computing resources were provided by the Climate Simulation Laboratory at NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation and other agencies.
A wave pouch is a region of approximately closed Lagrangian recirculation within the wave critical layer. It provides a favorable environment for vorticity aggregation and protects a protovortex within from the generally hostile tropical environment until the vortex is strengthened into a self-sustaining entity and emerges from the parent wave as a tropical storm.
The negative values in the left half of the x axis indicate downward motion. For example, it shows about 99% of the downward motion at 8-km altitude is weaker than 1 m s−1.
The contours in Fig. 3 and subsequent figures represent the frequency of vertical motion weighted by some quantity, such as vertical mass flux or vertical moisture flux. For example, the CCFADs of vertical mass fluxes indicate how upward (downward) motion less than or equal to the indicated value contributes to the total upward (downward) mass flux. Compared to the CCFAD of w (shading in Fig. 3), the weighting by the vertical mass flux decreases the CCFAD values at weak w (i.e., smaller contribution by weak w).
Note that the mean diameter is different from the diameter of the mean area (, where is the mean area). The diameter of the mean area in the HR (CR) run is about 3/7 of (2.5 times as large as) that in the control run.