A Numerical Investigation of a Supercell Tornado: Genesis and Vorticity Budget

Transcription

1 Journal of the Meteorological Society of Japan, Vol. 88, No. 2, pp , DOI: /jmsj A Numerical Investigation of a Supercell Tornado: Genesis and Vorticity Budget Akira T. NODA Research Institute for Global Change, Japan Agency for Marine-Earth Science and Technology, Kanazawa, Japan and Hiroshi NIINO Ocean Research Institute, The University of Tokyo, Tokyo, Japan (Manuscript received 2 June 2009, in final form 10 December 2010) Abstract The mechanism of supercell tornadogenesis and its vorticity budget are investigated by means of a highresolution (horizontally uniform grid size of 70 m) numerical simulation of the Del City storm, which occurred in Oklahoma, USA, on May 20, After 50 min of the simulation, a meso-low, which is generated by nonlinear interaction between the storm updraft and vertical wind shear associated with both environmental and storm-induced horizontal flow, develops at around 1.8 km above ground level (AGL). The meso-low acts to strengthen the underlying updraft, and generates a low-level mesocyclone via the tilting of horizontal vorticity associated with the environmental wind and that generated by baroclinic processes. In turn, a pressure depression associated with this low-level mesocyclone generates an updraft exceeding 43 m s 1 at 1.5 km AGL. In addition, small-scale vortices (pretornadic vortices) develop along a gust front. When the low-level updraft strengthened, one of these pretornadic vortices located immediately beneath the updraft shows a rapid growth into a major tornado. As the tornado develops, a downward pressure gradient force associated with an intense rotation of the tornado strengthens a tornado-scale downdraft on its north side. The developed downdraft compresses the vertical vortex of the tornado, eventually resulting in its dissipation. We also performed a vorticity budget analysis along a typical air-parcel trajectory. Air parcels in the mature tornado vortex originate mainly from the northwest in a layer between 10 and 500 m AGL. Vertical vorticity within the mature tornado is initially produced during descent via tilting of the horizontal vorticity, which is enhanced by stretching of the vortex tube. 1. Introduction Tornadoes are undoubtedly the most violent type of atmospheric vortex. Their structure and genesis are of great interest from the viewpoints of meteorology and fluid mechanics, as well as disaster mitigation. There are currently two accepted mechanisms of tornadogenesis: one associated with a Corresponding author: Akira T. Noda, Research Institute for Global Change, Japan Agency for Marine- Earth Science and Technology, Showamachi, Kanazawa, Yokohama City, Kanagawa , Japan. a_noda@jamstec.go.jp , Meteorological Society of Japan meso-scale front, and the other accompanying a supercell. The former mechanism appears to be relatively well understood based on detailed observational studies (e.g., Wakimoto and Wilson 1989) and high-resolution numerical simulations (e.g., Lee and Wilhelmson 1997). In contrast, the genesis mechanism of supercell tornadoes remains poorly understood despite extensive research e orts (e.g., Doswell and Burgess 1993; Burgess 1997; Wakimoto and Cai 2000). In recent decades, several studies have succeeded in undertaking close-range observations of a supercell tornado using innovative mobile Doppler radar systems, thereby revealing the characteristics of the

2 136 Journal of the Meteorological Society of Japan Vol. 88, No. 2 flow structure within such tornadoes (e.g., Bluestein et al. 2003a, 2003b, 2007; Alexander and Wurman 2005; Wurman and Alexander 2005). Bluestein et al. (2003b) reported an especially interesting flow field within a supercell tornado that developed at Bassett, Nebraska, USA, on July 5, Their results indicated that a leading edge of the bowshaped echo pattern developed ahead of the storm prior to tornadogenesis. The tornado developed around the echo region, and small-scale vortices developed around the tornado. In recent decades, numerical simulations also have shown great promise as a tool for understanding the overall process of tornadogenesis. Wicker and Wilhelmson (1995) (hereafter WW95) undertook simulations using a two-way nesting technique with outer and inner horizontal grid sizes of 600 and 120 m, respectively, and obtained a tornado vortex with a vertical vorticity of 0.31 s 1. They investigated the vertical structure and evolution of a mesocyclone and associated tornadoes. In addition, Grasso and Cotton (1995) used a two-way nesting technique for triply nested computational domains to investigate the spatial characteristics of a simulated low-level mesocyclone and updraft. Despite these previous e orts, several points remain unresolved, including details of the generation process (e.g., the triggering mechanism for the tornadogenesis and its relationship with the evolution of lowlevel mesocyclones) and the way in which the structure of a tornado vortex changes when interacting with the low-level mesocyclone. Numerical simulations have been performed in several studies with the aim of understanding the vortex structure of tornadoes (e.g., Fiedler and Rotunno 1986; Lewellen et al. 1997; Nolan and Farrell 1999); however, in most of these studies, an axisymmetric model has been employed that does not consider complex environmental structure associated with the parent storm. On the basis of mobile Doppler radar observations, Wurman and Gill (2000) analyzed the vortex structure of a supercell tornado that developed in Dimmit, Texas, USA, on June 2, Their results revealed the asymmetry of the tornado vortex, which cannot be explained solely in terms of surface frictional e ects. These observations indicate that the parent storm and/or lowlevel mesocyclone strongly modulate the vortex structure of tornadoes; consequently, it is necessary to take into account such interactions in order to understand the detailed evolution of tornadoes. To overcome the uncertainty regarding the nesting technique, for which artificial numerical di usions are sometimes required to reduce numerical noises, and to examine the temporal behavior of the parent supercell storm and associated tornado throughout the entire model integration-time, Noda and Niino (2005; hereafter NN05) successfully simulated a supercell tornado for the Del City storm that occurred in Oklahoma, USA (May 20, 1977), using a uniform horizontal grid size of 70 m. Their analysis demonstrated the importance of small-scale vortices for tornadogenesis (pretornadic vortices; one of which developed into a major tornado) along the gust front. For the Del City storm, both the simulated pretornadic vortices near the tornado and the spatial evolution of the gust front are surprisingly similar to, for example, those observed for a supercell tornado in Nebraska on June 5, 1999 (Bluestein et al. 2003a, 2003b). NN05 indicated that once a low-level mesocyclone has developed, one of the pretornadic vortices starts to grow and thereby develops into a tornado. In addition, the authors examined the structure of the tornado vortex in its mature phase. NN05 left several interesting issues to be investigated, including a detailed analysis of the mechanism of the rapid evolution of low-level mesocyclones and tornadoes, and a budget analysis of vorticity for a mature tornado. Accordingly, the aims of the present study are to (1) examine changes in the parent supercell storm prior to tornadogenesis, (2) explore temporal changes in the structure of the simulated tornado vortex as it interacts with the low-level mesocyclone, and (3) clarify the origin and evolution process of tornado vorticity during its mature phase via a vorticity budget analysis for backward trajectories of air parcels. 2. Model setup 2.1 Model description The model used in the present study is ARPS Version (Xue et al. 1995), which solves a quasi-compressible equation system. The 1.5-order turbulent closure scheme proposed by Deardor (1980) is used for computation of the eddy viscosity coe cient, and we employed the simple warm rain microphysics by Kessler (1969). As discussed below (Section 4.2), the tornadic supercell simulated in this study is primarily controlled by dynamical effects (e.g., vertical wind shear and related pressure deviation). Consequently, the inclusion of a cold rain microphysics would not alter the fundamental mechanism of tornadogenesis in the present case.

3 April 2010 A. T. NODA and H. NIINO The model domain size was determined on the basis of available computational resources. Though the vertical domain size of 15.1 km may be somewhat small for simulating a deep convective cloud, the role of the upper troposphere would not be of primary importance in tornadogenesis. In fact, the characteristics of the simulated storm are consistent with those commonly observed in typical supercell storms. 2 It is expected that surface friction strongly influences the structure of the tornado vortex via the formation of a frictional boundary layer. Because the inclusion of bottom friction would strongly deform the basic wind profile over time, and because it would complicate any attempt at a detailed analysis of the storm s evolution, we chose to exclude the e ect of bottom friction in the present analysis. The present approach is advantageous when the results are compared with those of the previous numerical studies on the Del City storm (e.g., Klemp and Rotunno 1983; Adlerman et al. 1999; Adlerman and Droegemeiner 2001; Noda and Niino 2003), most of which employed warm-rain microphysics. The influence of ice-phase condensates on tornadic supercells is another interesting topic for future study (cf. Johnson et al. 1993). The horizontal grid size (70 m) is uniform over the entire domain, while the vertical grid size is stretched from 10 m near the ground to 760 m near the domain top. The size of the computational domain is 66:4 66:4 km horizontally and 15.1 km vertically1. The bottom and lateral boundary conditions are free slip2 and open, respectively. A rigid wall is placed at the upper boundary, and a sponge layer (in which the e-folding time is 300 s) is placed above 12 km above ground level (AGL) to prevent the reflection of gravity waves from the top boundary. The earth s rotation is not considered. The time integration is performed for 3 h with time steps of 0.03 s and 0.18 s for acoustic and nonacoustic modes, respectively. 2.2 Initial environment The Del City storm (e.g., Klemp et al. 1981) is one of the most interesting cases for investigating tornadogenesis because its basic behavior has been extensively studied using relatively low-resolution models with a horizontal grid size of around 500 m (e.g., Rotunno and Klemp 1982, hereafter RK82; Adlerman et al. 1999). The setting adopted in the present simulation follows that employed in these past studies. One-dimensional sounding data (Figs. 1a, b), which represent a composite of the 1500 UTC Fort Sill (Oklahoma) and 1620 UTC Elmore City (Oklahoma) soundings, are given as a horizontally uniform initial environment, and an ellipsoidal thermal bubble (maximum amplitude of 4 K and horizontal and vertical diameters of 10 and 1.5 km, respectively) is introduced to initiate the storm. 3. Results Figure 2 shows the temporal evolution of the rainwater mixing ratio and horizontal wind vectors for the simulated supercell at 928 m AGL. After the storm is initiated by the thermal bubble (Fig. 2a), it develops rapidly and is split into left-moving (x ¼ 36:0 km and y ¼ 28:0 km) and right-moving storms (x ¼ 34:5 km and y ¼ 20:0 km). The rightmover develops more intensively and eventually evolves to the tornadic supercell. From here, we concentrate on the right-mover. A precipitating region becomes elongated in a north south orientation in response to di erential advection of the vertically sheared environmental flow (Fig. 2a; see also Fig. 1b). A region of heavy precipitation is found in the southeast, where the dominant convective activity occurs. After 2100 s (Fig. 2b), a hook-shaped rainwater distribution (as commonly observed for supercells by radar measurements) develops in the southeast part of the storm. This storm-scale feature becomes increasingly pronounced over time. After 4600 s (Fig. 2c), the storm increases in the horizontal dimension to more than 20 km, and a tornado develops at the tip of the hook (see Section 4 for details). Figure 3 shows a three-dimensional image of the simulated tornado viewed from the southeast. A wall cloud develops below the mean level of the cloud base, and a funnel cloud extends from the bottom of the wall cloud to the ground. In the funnel cloud, a column-like region of strong vertical vorticity extends from the ground into the wall cloud. The maximum horizontal wind speed (> 40 m s 1 ) is located near the tornado center on the southeast side. Figure 4 shows a close-up view of the wind and vertical vorticity fields over the region indicated by the thick square in Fig. 2c. A strong updraft, extending over a distance of 5 km, develops in a northwest southeast orientation by 3900 s (Fig. 4a). An intense downdraft in the southwest corner is formed in association with strong precipitation. A westerly flow that originated from the downdraft converges with the easterly environmental flow while wrapping up around the mesocyclone center at around x ¼ 29:2 km and y ¼ 21:9 km (Fig. 4b).

4 138 Journal of the Meteorological Society of Japan Vol. 88, No. 2 Fig. 1. Initial environment used in the present simulation of the Del City supercell storm of May 20, (a) Skew-T diagram. Solid and dashed lines show temperature and dew point temperature, respectively. The left-hand vertical axis shows temperature (degrees) and pressure (hpa), and that on the right shows the mixing ratio of water vapor (g kg 1 ). (b) Wind hodograph. The vertical and horizontal axes show the x- and y-components of the horizontal wind velocity, respectively. The term sfc near the solid square denotes the value near the ground, and the numbers indicate height (in kilometers) above ground level. The white open circle denotes the simulated storm motion estimated from movement of the mesocyclone center. Fig. 2. Temporal evolution of the supercell at 928 m AGL at times of (a) 1800 s, (b) 2100 s, and (c) 4600 s. Shading shows the rainwater mixing ratio (g kg 1 ) and arrows denote horizontal wind vectors (m s 1 ) for each block of 40 grid points. The square box in (c) indicates the region shown in Fig. 4.

5 April 2010 A. T. NODA and H. NIINO 139 Fig. 3. Three-dimensional view of the simulated tornado and the lower part of the storm, viewed from the southeast at 4504 s. The gray and red isosurfaces represent a cloud water mixing ratio of 0.1 g kg 1 and vertical vorticity of 0.6 s 1, respectively. The colored horizontal cross-section (the legend is provided in the upper right) shows wind speed at the lowest grid level (5 m AGL). The figure was constructed by clipping o the simulated data around the tornado. By 4504 s, a strong downdraft region gradually intrudes northward into the updraft core at x ¼ 28:7 km and y ¼ 22:7 km (Figs. 4c, e). Klemp and Rotunno (1983) performed a numerical simulation for the present storm under a relatively coarse grid resolution, revealing that low pressure evolved via strong horizontal rotational flow near the mesocyclone center. The downward nonhydrostatic pressure gradient force (PGF) exerted by the low pressure drives a dynamically induced downdraft, which Klemp and Rotunno termed occlusion downdraft. The downdraft observed in Figs. 4c, e is interpreted to represent the occlusion downdraft. At 4504 s, a region with strong vertical vorticity (> 0.5 s 1 ) exists at around x ¼ 29:3 km and y ¼ 23:4 km, indicating the development of a tornado. Another downdraft (x ¼ 29:3 km and y ¼ 23:5 km) develops on the northeast side of the tornado (described in detail in Section 4), induced by a downward PGF associated with the tornado itself. This downdraft is herein referred to as a tornado-induced downdraft. After 4302 s, vertical vorticity shows a dispersed distribution, indicating that the mesocyclone is losing its well-organized structure (Figs. 4d, f ). After 4700 s, when the tornado eventually dissipates, the downdraft divides the updraft core into southern and northern parts (Figs. 4g, h). The vertical vorticity of the mesocyclone, which has been maintained until now, begins to lose its organized structure and becomes dispersed. The mesocyclone eventually dissipates, evolving into a patchy distribution of vertical vorticity.

6 140 Journal of the Meteorological Society of Japan Vol. 88, No. 2 Fig. 4. Temporal evolution of the simulated field around the mesocyclone at 715 m AGL from 3900 s to 4700 s. (a) Vertical velocity (shading; m s 1 ) and (b) vertical vorticity (shading; s 1 ) and horizontal wind vectors (arrows for each block of 15 grid points; m s 1 ) for 3900 s. The thick solid contour lines are drawn for a vertical vorticity of 0.05 s 1 or larger with an interval of 0.1 s 1. (c) and (d), (e) and (f ), and (g) and (h) are the same as (a) and (b), except for 4302, 4504, and 4700 s, respectively. 4. Tornadogenesis 4.1 Formation of a mid-level meso-low NN05 analyzed the temporal evolution of minimum perturbation pressure, maximum vertical velocity, and maximum vertical vorticity in the simulated supercell storm (Fig. 5), where a perturbation of a physical quantity is defined by a deviation from the horizontally uniform initial state. The authors reported that a minimum perturbation pressure (Fig. 5c) began to develop at around 1800 m AGL after 3500 s, more than 20 min prior to initiation of the tornado (hereafter we refer to this midlevel pressure depression as a meso-low). This event was in fact a prelude to tornadogenesis. Subsequently, another perturbation pressure minimum developed rapidly at around 800 m AGL after 4100 s. This minimum corresponds to the evolution of the low-level mesocyclone, acting to accelerate the underlying updraft via upward PGF (Fig. 5a). The minimum eventually produces a low-level updraft stronger than 43 m s 1 at 1517 m. This rapid intensification of the low-level updraft is essential to tornadogenesis because it strengthens the vertical vorticity immediately below it via stretching (Fig. 5b). Here, we investigate how the meso-low is produced. Figure 6a shows a horizontal cross-section of the rainwater mixing ratio and pressure perturbation at 1903 m AGL. A significant pressure depression associated with the meso-low (characterized by the 3 hpa contour) occurs at around x ¼ 29:5 km and y ¼ 24 km. Figures 6b and 6c show vertical cross-sections of the vertical PGF and vertical velocity fields, respectively, through the center of the meso-low along the dash dotted line in Fig. 6a. The vertical PGF is directed upward below 1800 m AGL, and acts to accelerate the updraft at around 700 m AGL to > 5ms 1 (Fig. 6c). To clarify the formation mechanism of the mesolow, we introduce a diagnostic equation for the Exner function ðpþ under the Boussinesq approximation (e.g., RK82):

7 April 2010 A. T. NODA and H. NIINO 141 Fig. 5. Time height cross-section of the supercell from the start of the simulation. (a) maximum updraft, (b) maximum vertical vorticity, and (c) minimum perturbation pressure, at each height level. The contour lines for (a), (b), and (c) are drawn for each 5 m s 1, 0.2 s 1, and 1 hpa, respectively. The figure was constructed using data at 9 s intervals (after NN05).

8 142 Journal of the Meteorological Society of Japan Vol. 88, No. 2 Fig. 6. Storm-scale structure of the pressure field. (a) Horizontal cross-section of perturbation pressure and rainwater mixing ratio at 1.9 km AGL at 3300 s. Shading shows the rainwater mixing ratio (g kg 1 ) and dashed contour lines show the perturbation pressure (hpa) field. (b) Vertical cross-section of the perturbation pressure field and vertical PGF along the dash dotted line in (a) along y ¼ 24 km. Shading shows vertical PGF (10 1 ms 2 ) and dashed contour lines show the perturbation pressure. (c) It is the same as (b), except that vertical velocity (m s 1 ) is used instead of vertical PGF. " C p Y 0 2 p ¼ qb qz qu 2 þ qv 2 þ qw 2 qx qy qz # þ 2 qu qv 2 qu qw qy qx qz qx þ qv qw ; qz qy ð1þ where C p (¼ 1004 J K 1 ) is specific heat, B is buoyancy, and Y 0 (¼ 300 K) is the reference potential temperature. The first term on the right-hand side (RHS) of Eq. (1) is associated with buoyancy, and the other two terms are associated with dynamical e ects. The third term on the RHS represents the interaction between an updraft and vertical wind shear. The second term consists of the three terms representing the convergence and a term representing the vertical vorticity. Below, we show that the third term plays the most crucial role in terms of appearance of the meso-low considered in the present study. Figure 7 compares the magnitude of each term in Eq. (1) along the same vertical cross-section as that shown in Fig. 6a. The sum of the RHS terms (Fig. 7a) shows that a region of positive RHS is systematically distributed between x ¼ 27 km and x ¼ 33 km over km AGL; this causes the low pressure at around 1800 m AGL (note that the positive sign of the RHS contributes to negative p). Figures 7b d show that the third term is the primary contributor to the meso-low. Figure 8 shows the temporal evolution of horizontal wind vectors (Figs. 8a, c, e) and vertical shear vectors (Figs. 8b, d, f ) from 2700 s to 3600 s at 1800 m AGL, where the center of the meso-low appeared. The directions of the wind vectors become aligned over time and the storm-scale airflow becomes organized (Figs. 8a, c, e). Following this change in the wind vectors, the directions of the vertical shear vectors, which are initially somewhat disorganized (Fig. 8b), also become systematically organized (Figs. 8d, f ). For reference, the vertical wind-shear vector for the initial environment at the same altitude is shown in the upper right of Fig. 8b. The magnitude of this vertical shear vector is much smaller than the magnitude of that caused by the evolution of the storm by 3600 s, indicating that nonlinear e ects associated with the storm-induced vertical shear are the dominant contributor to the pressure perturbation. Organization of the directions of the wind vectors and wind-shear vectors occurs mainly in the northeastern part of the storm-scale updraft (i.e., at around x ¼ 30 km and y ¼ 24 km in Figs. 8e,

9 April 2010 A. T. NODA and H. NIINO 143 Fig. 7. Magnitude of each term on the RHS of Eq. (1) at 3300 s in the same cross-section as that shown in Figs. 6b and 6c. (a) Sum of all terms, (b) first term, (c) second term, and (d) third term (10 4 s 2 ). Contour interval is s 2. f ). These remarkable changes in the wind field appear to occur as a result of interaction between storm-scale circulation and the environmental wind. The environmental wind relative to the storm is southeasterly at the level where the meso-low develops (see Fig. 1b). In the presence of the storm, the environmental flow curves cyclonically around the storm-scale updraft, and moves into the storm, resulting in a horizontally nearly homogeneous wind field with vertical shear in the northeast of the storm-scale updraft, when the flow directions of the cyclonic circulation associated with the updraft are almost coincident with that of the environmental flow in this area. The wind field then produces the pressure depression owing to the mechanism suggested by RK82. On the other hand, along the southern and western sides of the stormscale updraft, the environmental flow from the southeast moves against the cyclonic flow associated with the rotating updraft. This leads to a region of nearly stagnant and disturbed wind field in the southwestern part of the storm-scale updraft; consequently, no systematic pressure perturbation is observed in this area. We also analyzed the development mechanism of the low-level mesocyclone as indicated by the

10 144 Journal of the Meteorological Society of Japan Vol. 88, No. 2 Fig. 8. Temporal evolution of vertical velocity, horizontal wind vectors, perturbation pressure, and vertical shear vector around the meso-low at 1.9 km AGL. (a) Vertical velocity (shading; m s 1 ) and horizontal wind vectors (arrows; m s 1 ) at 2700 s, (b) perturbation pressure (shading; hpa) and vertical shear vectors (s 1 ) at 2700 s. (c) and (d), and (e) and (f ) are the same as (a) and (b) except for 3300 and 3600 s, respectively. Vectors are drawn at every 20 grid points. The box in the upper-right corner of (b) shows the vertical shear vector (s 1 ) of the undisturbed field at 1.9 km AGL. pressure perturbation that appeared at around 800 m AGL (Fig. 5c). Our finding in this regard is similar to the explanation provided by Rotunno and Klemp (1985): the updraft accelerated by the meso-low acts to tilt upward the adjacent horizontal vorticity, which is intensified by baroclinicity near the gust front; subsequently, the updraft stretches it upward. 4.2 Low-level mesocyclone and tornado This subsection examines how the tornado develops as it interacts with the low-level mesocyclone, attains its mature phase, and eventually dissipates. To this end, we introduce the vertical momentum equation for an inviscid Boussinesq atmosphere: qw qt ¼ ~v ~ h w w qw qz qp0 qz þ B: ð2þ The first and the second terms of the RHS represent horizontal and vertical advection, respectively. The third term shows the vertical PGF, where p 0 is perturbation pressure, and the fourth term represents buoyancy. a. Developing phase Figure 9 shows a horizontal cross-section of the terms in the RHS of Eq. (2) around the low-level mesocyclone and the developing tornado (4170 s) at 500 m AGL. The figure also shows the tendency term of vertical velocity (left-hand side of Eq. 2), which is estimated using data stored every 2.88 s; the data are also used for budget analysis along air trajectories (see Section 5). Although these two quantities (sum of the RHS terms and the tendency term of Eq. 2) show some di erences, this does not

11 April 2010 A. T. NODA and H. NIINO 145 Fig. 9. Horizontal cross-sections of terms in the vertical momentum equation for the area near the tornado and low-level mesocyclone at 500 m AGL at 4170 s. (a) Vertical velocity (thin contour lines with shading; ms 1 ), (b) time rate of change of vertical velocity, (c) sum of the RHS terms in Eq. (2), (d) horizontal advection term, (e) vertical advection term, (f ) vertical PGF, and (g) buoyancy. The thin contour lines in (b) (g) are drawn for 0, G0:1, and G0:2 ms 2, and at 0.4 m s 2 intervals; shading indicates positive values. The thick contour lines in each figure show vertical vorticity at 0.1 s 1 intervals with zero lines omitted. The grayscales for (a) and for (b) (g) are shown below panel (a) and on the right side of panel (g), respectively. a ect the validity of the following results. Note that the discrepancies may be due to the di usion term and/or the crude nature of the estimate of the tendency using a time interval of 2.88 s, which is about 16 times larger than the time step for the time integration (0.18 s). Figure 9a shows two major regions with vertical vorticity greater than 0.1 s 1 at 500 m AGL: a region at around x ¼ 28:7 km and y ¼ 22:6 km corresponds to the low-level mesocyclone, and a second region at x ¼ 29:4 km and y ¼ 22:4 km corresponds to the tornado. A region of strong vertical velocity (> 30 m s 1 ) is associated with the low-level mesocyclone, and is located about 800 m west of the tornado. The sum of the RHS terms (Fig. 9c) is positive in most of the southern part of the tornado vortex. Among these terms, vertical PGF (Fig. 9f ) is dominant and is positive throughout almost the entire region. The horizontal advection term (Fig. 9d) is strongly positive (> 0.8 m s 2 ) in some regions, mainly on the south-to-east side of the low-level mesocyclone (x ¼ 28:8 km and y ¼ 22:5 km) and the tornado (x ¼ 29:5 km and y ¼ 22:3 km). These two regions with positive horizontal advection developed in response to eastward advection of intense updrafts in the west (Figs. 9a, d). The vertical advection term (Fig. 9e) and buoyancy (Fig. 9g) are mainly negative or weakly positive, indicating that neither plays a primary role in intensifying the updraft around the tornado. Figure 10 shows the same quantities as those in Fig. 9, in the vertical cross-sections through the low-level mesocyclone and the tornado at 4170 s; the time rate of change of vertical velocity is not shown hereafter. The vertical vorticity of the lowlevel mesocyclone is collocated with the intense updraft (> 30 m s 1 ) at around x ¼ 28:8 km below 700 m AGL (Fig. 10a). To the east of the mesocyclone below 700 m AGL, a region with significant

12 146 Journal of the Meteorological Society of Japan Vol. 88, No. 2 Fig. 10. Vertical cross-sections through the tornado center at y ¼ 22:5 km at 4170 s. (a) Vertical velocity (thin contour lines with shading; m s 1 ), (b) sum of the RHS terms of Eq. (2), (c) horizontal advection term, (d) vertical advection term, (e) vertical PGF, and (f ) buoyancy. The thin contour lines in (b) (g) are drawn for 0 and G0:05 m s 2, and at 1 m s 2 intervals; shading indicates positive values. The thick contour lines in each figure show vertical vorticity at 0.1 s 1 intervals with zero lines omitted. The grayscales for (a) and for (b) (g) are shown below panel (a) and on the right side of panel (g), respectively. positive vertical vorticity occurs at x ¼ 29:5 km, associated with the developing tornado. The sum of the RHS terms near the ground (< 400 m AGL; Fig. 10b) is positive on the west side and upper left corner of the tornado vortex, which acts to build up the tornado toward the lowlevel mesocyclone. The horizontal advection term (Fig. 10c) shows a less systematic distribution below 600 m AGL. The vertical advection term (Fig. 10d) is negative throughout almost the entire region, reflecting the fact that vertical velocity increases with height in and around the tornado vortex (Fig. 9a). Buoyancy is negative below 1 km AGL because of the cold air outflow produced by the storm. Overall, the RHS terms (Figs. 10b f ) reveal that vertical PGF makes the greatest contribution to acceleration of the updraft. b. Mature phase Figure 11 shows vertical cross-sections of the same quantities as those in Fig. 10, although during the mature phase of the tornado (4504 s). The updraft associated with the low-level mesocyclone (Fig. 11a) has clearly weakened as compared with the updraft at 4170 s (compare the shading in Figs. 10a, 11a corresponding to > 30 m s 1 ). The updraft region remains dominant in the tornado vortex center, but a downdraft has begun to develop in the upper right region of the tornado (e.g., x ¼ 29:7 km at 1 km AGL). A tornado-induced downdraft has begun developing in the north of the tornado vortex (see Figs. 4e, 13b), although it is not apparent in the vertical cross-section through the tornado center. In fact, the downdraft plays an important role in changing the horizontal advection term (see below). The vertical vorticity of the tornado has now reached the region of the intense updraft in the low-level mesocyclone (x ¼ 28:8 km at 800 m AGL). The sum of the RHS terms (Fig. 11b) is mainly negative along the axis of the tornado vortex, indicating that the updraft in the tornado vortex is

13 April 2010 A. T. NODA and H. NIINO 147 Fig. 11. It is the same as Fig. 10, except along y ¼ 23:3 km at 4504 s. weakening sharply. This intense downward acceleration of the vertical velocity prohibits the further evolution of the tornado; consequently, it starts to weaken via vortex compression. The vertical advection term (Fig. 11d) and buoyancy term (Fig. 11f ), the magnitudes of which are both less than 1.0 m s 2, are not the primary contributors to the trend in vertical velocity of the tornado vortex. A pair of positive and negative vertical PGFs exists on the western and eastern sides of the tornado vortex (Fig. 11e), reflecting the westward tilt of the tornado with increasing height. The horizontal advection term (Fig. 11c) shows a pattern similar to that of vertical PGF, although with opposite sign. This pattern is caused by cyclonic circulation, which advects the updraft and downdraft in the southern and northern parts of the tornado, respectively, as described below (see Fig. 13b). We also examined the sum of vertical PGF and the horizontal advection term in the tornado vortex (not shown). The result largely explains the negative tendency (sum of the RHS terms; Fig. 11b) along the tornado axis: near the vortex center, in the eastern half, the negative vertical PGF overwhelms the positive horizontal advection, whereas in the western half, the negative advection term dominates the positive vertical PGF (Figs. 11c, e). c. Dissipation phase Figure 12 shows vertical cross-sections for the dissipation phase of the tornado at 4703 s. The vertical velocity field (Fig. 12a) shows significant changes compared with the mature phase (4504 s; Fig. 11a): the vertical velocity accompanying the low-level mesocyclone weakens to less than 20 m s 1, and the downdraft occupies most of the region in the tornado vortex. Compared with earlier time points, the vortex has a much wider distribution of vertical vorticity. The sum of the RHS terms (Fig. 12b) shows negative values throughtout most of the tornado vortex, acting to further strengthen the downdraft. The tornado decays due to vortex compression. Vertical PGF (Fig. 12e) is positive below 500 m AGL due to the pressure depression located immediately above. However, the negative horizontal advection (Fig. 12c), which reflects the southward advection of the downdraft (see also Fig. 13e), now overcomes the positive vertical PGF in most

14 148 Journal of the Meteorological Society of Japan Vol. 88, No. 2 Fig. 12. It is the same as Fig. 10, except along y ¼ 23:2 km at 4700 s. regions (Figs. 12b, c, e). The vertical advection (Fig. 12d) and buoyancy (Fig. 12f ) combine to intensify the downdraft in the tornado core. In summary, vertical PGF generated by the lowlevel mesocyclone and tornado plays a primary role in changing the surrounding vertical velocity field, thereby controling the life cycle of the tornado. The horizontal advection term also plays an important role during the mature and dissipation phases when the vertical velocity around the tornado shows a strong non-axisymmetry. Vertical advection and buoyancy are of secondary importance in this regard. 5. Vortex structure and vorticity budget 5.1 Vortex structure Figure 13 shows the horizontal distributions of vertical vorticity and vertical velocity around the simulated tornado during the mature (4504 s) and dissipation (4607 s) phases. Although the vertical vorticity of the tornado during the mature phase (Fig. 13a) possesses a nearly axisymmetric structure, it also has a slight non-axisymmetric component, as indicated by the 0.15 s 1 vorticity contour; this component is presumably because of barotropic instability. The vertical velocity (Fig. 13b) shows that a downdraft region occupies most of the northern part of the vortex, while the southern part is dominated by an updraft region. Wind speed (Fig. 13c) shows a significant deviation from an axisymmetric structure: the highest wind speed (> 45 m s 1 ) occurs on the eastern side of the tornado vortex. During the dissipation phase, the downdraft in the vortex intensifies further (Fig. 13e), covering most of the vortex center at x ¼ 29:6 km and y ¼ 23:3 km. The wind speed field (Fig. 13f ) shows a more axisymmetric structure than that found during the mature phase, with a reduced maximum wind speed of less than 35 m s 1. Vertical vorticity is compressed by the downdraft at the vortex center, forming a donut-shaped pattern (Fig. 13d). Bluestein et al. (2003b) reported a similar expansion of the vortex of the Bassett tornado during its dissipation phase, noting that the diameter of the tornado vortex increased from 140 m to 180 m over a period of 20 s. The present tornado vortex increased in diameter from 400 m to 600 m (at 5 m AGL) over a period of 103 s. These results indicate that the morphology of a tornado vortex can

15 April 2010 A. T. NODA and H. NIINO 149 Fig. 13. Horizontal cross-section through the tornado at 5 m AGL after 4504 s. (a) Vertical vorticity at 0.1 s 1 intervals for values above 0.05 s 1, (b) vertical velocity at 0.2 m s 1 intervals, and (c) horizontal wind speed at 5 m s 1 intervals. The thick lines in (b) are isolines of vertical vorticity at 0.05 s 1. (d), (e), and (f ) are the same as (a), (b), and (c), respectively, except for 4607 s. change markedly over a period of just several tens of seconds. 5.2 Backward parcel trajectories of the mature tornado To understand the formation process and vorticity source of the mature tornado, here, we explore the origin of the airflow that constitutes the tornado vortex on the basis of analyses of 10-min backward trajectories of air parcels distributed in the tornado vortex. The backward trajectories are computed from the x-, y-, and z-components of the wind velocity stored on a hard disk at 2.88 s intervals. We chose this time interval based on a trial-and-error approach that indicated it yielded su cient accuracy. The location of a parcel 2.88 s before the present parcel is obtained using the wind velocity linearly interpolated from the surrounding eight grid points. The backward trajectory for each air parcel is obtained by repeating this procedure 208 times (i.e., s). Figure 14a shows the 10-min backward trajectories of 30 air parcels distributed at approximately equal spacing along the 0.2 s 1 isoline of vertical vorticity for the mature tornado vortex at 85 m AGL and 4504 s. There are two main airflow paths that enter the tornado vortex at 85 m AGL: one originates from low levels to the northeast of the tornado, travels southward below 10 m AGL, and flows into the tornado along an approximately linear path (Path 1); the other originates from a layer at m AGL to the northwest of the tornado, curves cyclonically as it descends, and flows into the tornado near the ground (Path 2). Twenty-nine of the 30 analyzed trajectories belong to Path 2. We also analyzed the backward trajectories of 30 air parcels placed along the 0.2 s 1 isoline of vertical vorticity at 5, 50, and 100 m AGL, revealing that 30, 28, and 30 air parcels followed Path 2, respectively (not shown). We therefore conclude that the dominant fraction of airflow that reaches the tornado arrives via Path Vorticity budget along a parcel trajectory Here, we study the vorticity budget along a typical air parcel that travels along Path 2. This parcel is hereafter referred to as Parcel A (see Fig. 14). The vorticity equations under the Boussinesq ap-

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17 April 2010 A. T. NODA and H. NIINO 151 Fig. 15. Temporal variations in the height of Parcel A. proximation for the horizontal and vertical components, ~o h and v, are respectively written as d~o h ¼ð~o ~ Þ~v h þ ~ ~B þ ~F Dh ; ð3aþ dt dz dt ¼ð~o ~ Þw þ F Dz ; ð3bþ where ~o ð¼ ðx; h; zþþ is the vorticity vector, ~B ð¼ ð0; 0; BÞÞ is buoyancy, and ~v h is the horizontal velocity vector. ~F Dh and F Dz are di usion terms that are not considered in the followingp analysis. An equation for the magnitude of o h ð¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ h 2 Þ is obtained by operating an inner product of ~o h on both sides of Eq. (3a) as follows: do h ¼ 1 doh 2 dt 2o h dt ¼ 1 x h qu o h qy þ z qu þ h z qv qz qz þ x qv qx þ 1 x 2 qu qv þ h2 o h qx qy þ 1 x qb o h qx þ h qb : ð4aþ qy Equation (3b) can also be rewritten as dz dt ¼ x qw qx þ h qw þ z qw qy qz : ð4bþ The first term on the RHS of Eq. (4b) represents vorticity tilting, and the second term represents stretching. Similarly, the first term on the RHS of Eq. (4a) expresses tilting; the second term, stretching; and the third term, baroclinic production. It is important to consider the proposal by Davies-Jones (1982) that the tilting term not only reorients the vortex tube but also stretches it. Although we note that the first term on the RHS of Eq. (4a) does not vanish in general, we follow the conventional usage of terms as described above. Figure 15 shows temporal variations in the height of Parcel A. The parcel is located to the northwest of the tornado vortex at 370 m AGL at 3904 s, and descends as it approaches the tornado, reaching 50 m AGL at 4440 s. The parcel then moves almost horizontally, entering the tornado from the east through its updraft edge before rising to 85 m AGL at 4504 s in response to the updraft located near the tornado. g Fig. 14. (a) Backward trajectories calculated for 30 air parcels distributed at approximately equal spacing along the vorticity contour line of 0.2 s 1 around the tornado (see text for details). Contour lines indicate the vertical velocity at 85 m AGL for 4504 s. The initial locations of the 30 air parcels are shown in the inset at lower right, with vertical vorticity contoured at 0.1 s 1 intervals. The heights of the parcels are indicated according to the color scheme outlined in the upper left of the figure. (b) Trajectory of Parcel A, along which the vorticity budget is analyzed. The arrows superimposed on the trajectory show the horizontal components of the Lagrangian vorticity vector of Parcel A at each time.

18 152 Journal of the Meteorological Society of Japan Vol. 88, No. 2 Before performing a detailed analysis of the vorticity budget, we assessed the accuracy of the analysis. The results presented in the Appendix show that the temporal evolution of vorticity components, as obtained by integrating the tendency terms of the vorticity equations (Eqs. 3a and 3b) along the trajectory of Parcel A (Lagrangian vorticity), shows reasonable agreement with Eulerian vorticity, which is obtained by linearly interpolating the values of the eight grid points surrounding Parcel A at each time step. To help understand the relation between wind and vorticity vectors, we consider a normalized storm-relative helicity (NSRH; Lilly 1983): NSRH 1 ~o ~V j~ojj~vj ; ð5þ where ~V ¼ðu; v; wþ is a velocity vector relative to the storm motion. Figure 16 shows NSRH along the trajectory of Parcel A. NSRH at 3904 s is 0.81, meaning that Parcel A has a large streamwise vorticity component. NSRH increases with decreasing distance to the tornado, and approaches unity at 4200 s, maintaining this value until 4370 s, when it decreases to 0.97 before again approaching unity. Thus, the velocity and vorticity vectors for Parcel A are oriented in approximately the same direction over the course of the parcel s approach to the tornado (see also the vorticity vectors in Fig. 14b). Figures 17, 18 show values of o h and z, as well as their stretching and tilting terms, over the periods from 3904 s to 4355 s and from 4355 s to 4504 s, respectively. Also shown is a baroclinic term for o h. Note that the vertical scales in the two figures are di erent, as the magnitudes of the terms show a marked increase as the parcel approaches the tornado. Parcel A originally has a horizontal vorticity of s 1 at 370 m AGL at 3904 s (Fig. 15 and solid line in Fig. 17a). The horizontal vorticity gradually increases to by s 1 by 4270 s. The velocity vector and vorticity vector have approximately the same direction, as indicated by NSRH > 0:8 (Fig. 16). Figure 17a shows that the stretching term is always positive, making the dominant contribution to the increase in horizontal vorticity. The tilting term is largely secondary in this regard, but overcomes the stretching term between 4080 s and 4170 s, acting to strengthen the horizontal vorticity. The baroclinic term remains weakly positive until 4125 s, when it becomes weakly negative. Under the influence of the stretching and tilting terms, z varies at magnitudes less than 10 3 s 1 : positive values of z have yet to be generated for the tornado (Fig. 17b). In fact, z becomes negative at 4365 s and remains weakly negative until 4395 s. Horizontal vorticity, which slowly decreases after 4270 s, begins to increase again after 4385 s because of stretching (Figs. 17, 18), and remains stronger than s 1 until 4470 s when it shows a rapid increase (Fig. 18a and vorticity vectors in Fig. 16. Temporal variations in normalized storm-relative helicity (Eq. 5) along the trajectory of Parcel A.

19 April 2010 A. T. NODA and H. NIINO 153 Fig. 17. Vorticity budget along the trajectory of Parcel A between 3904 and 4355 s for (a) magnitude of horizontal vorticity o h and (b) vertical vorticity z. Solid, dashed, and dotted lines show the vorticity, stretching, and tilting terms, respectively. The dash dotted line in (a) shows the baroclinic term. The vertical axis on the left-hand side represents vorticity (s 1 ); that on the right-hand side represents the other terms (s 2 ). Fig. 14b). Parcel A enters the updraft region of about 0.2 m s 1 after 4435 s. The tilting term of z (Fig. 18b) becomes positive after 4435 s, producing the vertical vorticity of the tornado vortex. Parcel A begins to experience strong updraft after 4465 s at 300 m from the tornado center (Fig. 14b). Vertical stretching of vertical vorticity leads to a rapid increase in vertical vorticity to values above 0.3 s 1 (note that the scale of the vertical axis in Fig. 18b for the tilting term is 10 times smaller than that for the stretching term). The stretching term shows a steady increase until 4494 s, eventually exceeding 3: s 2 ; in response, vertical vorticity increases to 1: s 1 over the 14 s period from 4480 s to 4494 s. The horizontal vorticity decreases as the parcel approaches the tornado (@ 4495 s), because of the tilting term, but then starts to grow again after 4502 s by stretching. The baroclinic term contributes to the generation of horizontal vorticity in regions outside the tornado (approximately 380 s before Parcel A enters the tornado), but mainly until 4125 s (Figs. 17, 18). 6. Discussion The generation mechanism of mid-level mesolows in supercells and their dynamical role in determining storm behavior has been extensively studied from various perspectives. RK82 investigated the generation mechanism of a mid-level meso-low in a supercell storm, which is important in terms of the development of a right-moving supercell storm.

20 154 Journal of the Meteorological Society of Japan Vol. 88, No. 2 Fig. 18. It is the same as Fig. 17, but for the period from 4355 s to 4504 s. Note that the scales of the tilting term and the stretching term in (b) are di erent, being (10 4 s 2 ) and (10 3 s 2 ), respectively. Note that the scales for the vertical axes on the left (vorticity) and right (tendency terms) of the figure are di erent from those in Fig. 17. Using linear theory, the authors showed that the meso-low is generated via interaction between vertical wind shear of the environmental flow and the storm updraft (i.e., the third term on the RHS of Eq. 1), although they also noted that the nonlinear e ects might be important. The present results indicate that formation of the meso-low, which plays an important role in eventually generating a strong low-level updraft that leads to tornadogenesis, is induced by a nonlinear interaction between storm-induced vertical wind shear and the storm updraft. An interesting topic for a future study is to investigate why the storm-induced flow is organized so as to align the vertical shear vectors, thereby generating a significant meso-low, and to examine what mechanisms control the timing of development of the meso-low. On the basis of the results of our present and former studies, we propose several speculative suggestions that may help to further our understanding of tornadogenesis. NN05 reported that several smallscale vortices (pretornadic vortices) are generated along the gust front, presumably because of shear instability. Most of the vortices grow to a certain amplitude before eventually dissipating. The characteristics of these small-scale vortices resemble those of gustnadoes, which are commonly observed prior to major tornadoes (e.g., LeMone and Doswell 1979). In an earlier simulation, WW95 reported the transient evolution of vertical vorticity 20 min prior to tornado evolution, and noted that such a vortex might correspond to a gustnado associated with a supercell. Our results suggest that these gustnado-like vortices may represent vortices

21 April 2010 A. T. NODA and H. NIINO 155 that fail to develop into a major tornado because their timing and location do not match those of the developing strong updraft associated with the low-level mesocyclone. Two types of tornadoes are currently recognized: one type associated with a meso-scale front (e.g., Wakimoto and Wilson 1989) and the other associated with a supercell. Our results indicate that the generation mechanisms of these two types of tornadoes may be similar in terms of the immediate source of vorticity, which is vertical vorticity in the horizontal shear flows that pass along a convergence zone near the ground. However, supercell tornadoes are generally more violent and long-lived than are non-supercell tornadoes. This observation possibly reflects the fact that in a supercell, strong updraft moves largely in tandem with a gust front as a single system; consequently, the tornado is maintained via vertical stretching over a long period while being provided with vorticity (or circulation) by the parent storm. Recent observational data reveal that the probability of tornado occurrence once a mesocyclone has been detected is only 20 40% (e.g., Burgess 1997), demonstrating the need for further knowledge regarding the detailed mechanism of tornadogenesis. Wakimoto and Cai (2000) reported the nearly identical structure of mesocyclones in tornadic and nontornadic supercells, suggesting that the existence of a mesocyclone alone is insu cient to guarantee tornadogenesis. Our numerical simulation of the supercell tornado suggests that tornadogenesis requires not only the development of a mesocyclone, but also the presence of a pretornadic vortex (or at least circulation) associated with the gust front, located immediately beneath the intense low-level updraft associated with the low-level mesocyclone. Our results of the backward trajectory analysis are similar in several respects to the findings of WW95, who also reported two major paths that flow into a simulated tornado vortex. WW95 performed a vorticity budget analysis along the trajectories of air parcels, revealing that the vorticity source for the tornado was baroclinically generated horizontal vorticity. The finding that air parcels of the tornado vortex move along two paths may be explained by the classical view of the typical airflow structure around a mesocyclone formed in a supercell storm (see the conceptual model proposed by Browning 1964). However, the main paths of air parcels obtained in the present study are di erent from those described by WW95, who showed similar numbers of air parcels that travel along each path. In contrast, we found that almost all of the air parcels originated from the layer between 10 and 500 m AGL at 10 min prior to development of the mature tornado (i.e., Path 2), with few originating from the strong baroclinic region (i.e., Path 1). Therefore, the baroclinic production of horizontal vorticity contributes relatively little to the tornado (dash dotted lines in Figs. 17a, 18a). The di erences between the present study and WW95 regarding the major paths into the developing tornado may arise from di erences in the environmental fields, thereby indicating that the vorticity source for a mature tornado may be strongly sensitive to the environmental field. Many studies have investigated tornadogenesis from observational and theoretical perspectives (e.g., Davies-Jones 2008; Byko et al. 2009; Sasaki 2009). Several recent studies have focused on the role of a rear flank downdraft (RFD; e.g., Markowski 2002). The present results show that most of the air parcels that constitute a mature tornado originate in the RFD region (see the 30 parcel trajectories in Fig. 14a), thereby demonstrating that RFD plays an important role in terms of the transport of vorticity into a mature tornado. In this regard, the present results also raise the possibility that the major source of vorticity supply di ers depending on the stage of the life cycle: pretornadic vortices are important in terms of initiating the tornado (NN05), whereas the supply of vorticity from the parent storm via storm-scale downdraft (i.e., RFD) is important once a tornado starts to evolve. In terms of the structure of a tornado vortex, several observational studies have reported structures of multiple vortices structures in supercell tornadoes, and have attempted to clarify the nature of their non-axisymmetric structures (e.g., Wurman and Gill 2000; Alexander and Wurman 2005). The tornado vortex simulated in the present study has a single-cell structure throughout its lifetime except in its dissipation phase, as indicated by the distribution of vertical vorticity (Fig. 13). Whereas the vertical vorticity and pressure fields are approximately axisymmetric, the horizontal and vertical wind fields show significant deviations from axisymmetry. NN05 explained that it is not inconsistent for an axisymmetric vorticity pattern and asymmetric vertical velocity pattern to coexist since stretching (compression) of vertical vorticity by the updraft (downdraft) is compensated by horizontal