1. Introduction
Vortex breakdown, a rather well-known phenomenon in fluid mechanics, has been documented in tornadoes (Pauley and Snow 1988; see also Lugt 1989). The existence of vortex breakdown in tornadic mesocyclones has been inferred from observations of a downdraft near the central axis of the mesocyclonic vortex (Brandes 1978; Wakimoto and Liu 1998); tornadogenesis was attributed by these authors to the purported mesocyclonic vortex breakdown. Numerical models, however, have simulated the formation of such a central or axial downdraft and, thus, of a two-celled vortex in the absence of vortex breakdown.
The purpose of this note is to (i) demonstrate that vortex breakdown likely is precluded in mesocyclones, and (ii) provide a simple reinterpretation of these past and recent observations of central downdrafts and, hence, of two-celled mesocyclonic vortices. These tasks are accomplished through a review of the literature and through a consideration of the observations and of analytic and numerical model solutions (section 2). A theorized tornadogenesis process in two-celled mesocyclones is recalled in section 3. Conclusions are given in section 4.
2. Two-celled vortices and vortex breakdown
a. Background
Swirl ratio (S) is the ratio of the tangential velocity at the outer edge of the updraft hole of a laboratory tornado chamber (or the atmospheric analogue thereof), to the average vertical velocity through such an updraft hole (Davies-Jones 1973). The dependence on swirl ratio of tornadic vortex structure has been demonstrated in tornado chambers (e.g., Ward 1972; Church et al. 1979) and numerical models thereof (e.g., Rotunno 1977; Howells et al. 1988). Indeed, the vortex flow undergoes a remarkable metamorphosis as S increases (e.g., see Snow 1982; Davies-Jones 1986). Briefly, small values of S support a laminar, one-celled vortex: air rises throughout the vortex. Moderate swirl ratios support an intense vortex, within which axial or vertical motions now take the form of a jet that erupts out of the boundary layer. The vertical jet terminates aloft at a stagnation point along the central axis, above which (i) the vortex core broadens substantially, (ii) the flow becomes turbulent, and (iii) recirculatory or downward motion resides along the axis (Fig. 1a): these are characteristics of the breakdown of a columnar vortex that terminates at a solid boundary (or end wall), in accordance with the accepted definition of vortex breakdown observed in nature and technology (Lugt 1989). With subsequent increases in S, the vortex breakdown point becomes positioned successively closer to the lower surface until ultimately, the vortex breakdown penetrates to the lower surface. This results in a broadly rotating, two-celled vortex with downflow along its central axis, terminating in low-level radial outflow that turns vertical in an annular updraft at an outer radius (Fig. 1b).
The dynamics attendant with a rotating flow above a rigid boundary (like the earth’s surface) is crucial to development of an axial jet within simulated vortex cores and, presumably, within tornadoes. Assume that the rotating flow is in cyclostrophic balance, well above the rigid boundary and, hence, also above the boundary layer. Within the boundary layer, the cyclostrophic balance condition is upset because friction reduces the tangential wind and centrifugal force to zero at the ground;friction does not alter the radial pressure gradient force, however. The consequential unbalanced inward pressure gradient drives a strong inflow that transports parcels (that nearly conserve angular momentum) much closer to the central axis than is possible without friction. At a radial distance on the order of the boundary layer depth, inward transport of mass and angular momentum ceases as the flow turns vertical, erupting into an intense axial jet (Burgraff et al. 1971); this rotating jet may break down under certain conditions.
Although vortex breakdown “is still considered a basic, largely unexplained phenomenon in modern fluid dynamics” (Rusak et al. 1998), its existence, as just mentioned, appears to be tied in some way to the existence of an upstream1 axial jet. Such a jet that exceeds the axial motions outside the vortex core, perhaps by a factor of ≥3 (Leibovich 1984), affords the condition of supercriticality. In a supercritical vortex, centrifugal waves cannot propagate upstream. This condition contrasts that of subcriticality that exists downstream of the breakdown point and allows for upstream and downstream propagation of centrifugal waves (Benjamin 1962). According to Leibovich (1984), the general concept of critical flow is the consistent element in “all formations of criteria for onset of breakdown in purely axisymmetric theories.”
Summarizing, a supercritical vortex and moderate-to-high swirl ratio2 are necessary for vortex breakdown. Supercriticality in tornadoes is provided through a boundary layer–erupting axial jet in the vortex core. We consider next if the mesocyclone dynamics supports such an axial jet and hence the condition of supercriticality.
b. Analytic models
With the reasonable assumption that the velocity distribution of a Rankine-combined vortex3 approximates that of a mesocyclone (and that of a tornado), the solution due to Bodewadt (1940) represents the effects of a low-level mesocyclone core and its boundary layer. Bodewadt examined a fluid that rotates as a solid-body above a fixed plate, and found that the resulting vertical velocity distribution is independent of the radial distance from the axis of rotation (see Schlichting 1979, 225–230). This result implies the existence, at the top of the boundary layer, of a uniform updraft distributed throughout the mesocyclone core (of a few-kilometer radius) rather than of a narrow vertical jet on the mesocyclone axis (see Rotunno 1986, p. 430). Such a vertical jet does appear, however, in solutions of Burgraff et al.’s (1971) model of a potential vortex above a fixed plate; their model represents reasonably well a narrow tornado and its boundary layer, since the presumed potential flow of the tornadic vortex is large compared to its core.
A swirling, boundary layer–erupting jet as in Burgraff et al.’s solution (and in some tornadoes) is much more likely to be supercritical than is a broadly rotating updraft as in Bodewadt’s solution (and hence in some low-level mesocyclone cores) (R. Davies-Jones 1999, personal communication). Thus, by the discussion in section 2a, it is correspondingly more unlikely for a mesocyclone to develop a vortex breakdown. Two-dimensional (2D), axisymmetric, numerical-model experiments now are used to illustrate the formation of an axial downdraft and, thus, of a two-celled vortex in the absence of vortex breakdown.
c. Numerical model results
The upper boundary (z = 1) and outer wall (r = 3) are rigid, impermeable, and no slip; the central axis (r = 0) is a symmetry axis; and the bottom boundary (z = 0) is rigid, impermeable, and can be made no slip or free slip. There is no viscous boundary layer (like the steady Ekman layer over the solid earth or the unsteady vortex boundary layer simulated herein) when the free-slip boundary condition is imposed. Thus, the two simulations presented below examine the behavior of a vortex, with and without a viscous boundary layer. The choice of boundary condition (hence presence or absence of a boundary layer) is motivated by the presumed relevance of the boundary layer to the scale of the vortex being considered, as suggested by the model solutions in section 2b: the boundary layer of a mesocyclone apparently has only a minor effect on the mesocyclone core dynamics, whereas the boundary layer of a tornado has a profound effect on the tornado core dynamics.
1) Free-slip lower boundary condition
When a free-slip lower boundary condition is imposed, the model produces a subcritical vortex that becomes two-celled in absence of a vortex breakdown (Fig. 2). Rapid development of an axial downdraft and consequential evolution toward the two-celled vortex solution can be explained as follows (see also Rotunno 1984): Radially inward (outward) transport of angular momentum (Γ = rυ) surfaces by the low-level (upper-level) branch of the buoyancy-forced meridional circulation results in angular momentum surfaces that slope outward with height (Rotunno 1977). Such radially outward-sloping Γ surfaces imply that tangential velocity and hence vertical vorticity [ζ = (1/r)∂/∂r(υr)] decrease with height; rotationally induced pressure that is lower near the ground than aloft is furthermore implied. An adverse vertical pressure gradient force results along the central axis, which through the vertical momentum equation drives the axial downflow. This process, incidentally, represents the choking or vortex valve effect discussed by Lewellen (1971) and later applied by Lemon et al. (1975).
A relevant atmospheric analogue can be identified. Mesocyclones form at midlevels within a storm through tilting and subsequent stretching of the horizontal vorticity associated with vertical shear of the ambient horizontal wind. At low levels, mesocyclones form through tilting of horizontal vorticity associated presumably with horizontal buoyancy gradients. Cloud model simulations (e.g., Klemp and Rotunno 1983) and observations (Cai and Wakimoto 1998) have shown that the resultant vertical vorticity in the low-level mesocyclone can become larger than that of the midlevel mesocyclone. As just explained, this vertical distribution of ζ and, hence, pressure affords axial downdraft formation and a transition to a two-celled vortex; two-celled mesocyclones have been simulated (Rotunno 1984; Klemp and Rotunno 1983), observed by Doppler radar (Brandes 1978; Wakimoto and Liu 1998; Trapp 1999), and documented visually (Bluestein 1985).
2) No-slip lower boundary condition
When a no-slip lower boundary condition is imposed, the model produces, relative to the free-slip case, an intense tornado-like vortex with a narrow core (Fig. 3);this result is consistent with modeling results presented by Rotunno (1977), Howells et al. (1988), Fiedler (1993), and others. Within the vortex core is a vertical jet with windspeeds that are more than twice those outside the core (Fig. 3c). At time t = 6, the vertical jet stagnates on the axis at a height of z = 0.3. Below this height, the vortex is supercritical. Above this height, the vortex is subcritical and exhibits the definitive breakdown features [e.g., according to Hall (1972), Leibovich (1978), Leibovich (1984), and Lugt (1989)] like an abruptly broadened core and axial downflow; the axial downflow is present due to the decrease with height of tangential velocity, as similarly explained previously.
It is important to note here that the axial downdrafts within the two-celled vortex in the free-slip case and downstream of vortex breakdown in the no-slip case both are associated with adverse vertical pressure gradients. Moreover, the adverse pressure gradients are, in turn, associated with rotation that decreases in strength with height. A distinction can be made, however, between the two processes that result in this vertical distribution of ζ in a tornadic vortex undergoing breakdown (analogous to the no-slip case) and in a two-celled mesocyclone (analogous to the no-slip case). In the former, boundary layer processes act to amplify ambient (or mesocyclonic) ζ selectively at low levels, and in the latter, low-level ζ is generated in amounts greater than that at midlevels presumably by the vertical tilting of horizontal baroclinic vorticity, as discussed previously.
The two model solutions lead to the following important conclusion: the existence of a subcritical vortex with an axial downdraft does not imply that, at some previous time, the vortex necessarily had a breakdown and was supercritical. This conclusion is exemplified by the vortex in the free-slip case that was at all times subcritical, and, is used to help reinterpret the observations presented next.
d. Observations of two-celled mesocyclones
Recall that the existence of vortex breakdown in the tornadic mesocyclones of the 16 May 1995 Garden City, Kansas, tornadic supercell (Wakimoto and Liu 1998; Wakimoto et al. 1998) and the 8 June 1974 Harrah, Oklahoma, tornadic supercell (Brandes 1978) were inferred from observations of a downdraft near the central axis of the mesocyclonic vortex. In light of the preceding discussion, the interpretation of such mesocyclonic vortex breakdown is now evaluated.
Consider the two-celled mesocyclone in the Garden City supercell. Wakimoto and Liu retrieved the three-dimensional airflow of the storm from airborne Doppler radar data, and observed a central or occlusion downdraft (OD) within the storm’s mesocyclone. Wakimoto and Liu equated such axial downflow to that found downstream of a vortex breakdown: “It is likely that the breakdown reaches the surface (leading to a rapid expansion of the core radius) after the 2324–2329 volume time.”
The Garden City data indicate that if a hypothetical mesocyclonic vortex breakdown had formed and consequently reached the surface, it would have done so prior to the 2324–2329 UTC airborne radar scan: vertical cross-sections in Fig. 10 of Wakimoto et al. show weak-to-moderate downflow throughout the entire depth of the mesocyclone already at 2324–2329 UTC (just before tornadogenesis; their Fig. 10f) and even at 2319–2324 UTC (their Fig. 10e). If a mesocyclonic vortex breakdown had existed prior to these times, the data would show a marked axial flow transition aloft or near the ground, and at least the semblance of a boundary layer–erupting vertical jet or updraft core; the data afford the resolution of these features, based on the ∼2 km mesocyclone core radius at times preceding downdraft development. However, one instead finds at 2308–2315 UTC, upward vertical motion and positive vertical vorticity that are correlated through a 5 km depth, and, a vertical velocity field with a maximum at 5 km and small values in the lowest kilometer (their Fig. 10d). A similar vertical distribution of vertical velocity can be inferred at 2313–2318 UTC, from time–height profiles of vertical velocity and vertical vorticity (their Fig. 13). Hence, prior to the development of an axial downdraft and two-celled vortex, the data present no evidence of a supercritical vortex and associated breakdown. One may conclude, therefore, that the OD in the two-celled mesocyclone formed analogously to that in the vortex in the free-slip simulation.
Of course, one could argue that the hypothetical mesocyclonic vortex breakdown developed aloft and subsequently penetrated the boundary layer within 5 min, the approximate time difference between successive airborne volumetric radar scans. While this possibility cannot be ruled out, a dimensionalization of the no-slip model results suggests that it is unlikely. Consider a convective timescale of 7 min, which is derived from a convective velocity scale of 30 m s−1 and a length scale of 13 km; the velocity and length scales are inferred from Garden City storm data presented by Wakimoto et al. This timescale is used to dimensionalize the time difference between the incipient and well-developed vortex breakdown stages depicted in Figs. 3b and 3c, respectively. The resultant difference of 11.5 min implies that the radar data have sufficient temporal resolution to capture the development and evolution of a hypothetical breakdown.
3. Discussion
The two-celled mesocyclonic vortex is susceptible to a cylindrical vortex-sheet instability (Rotunno 1984). Once/if this instability is released, vortices (or vertical vorticity maxima; Fig. 7d, Klemp and Rotunno 1983) that are smaller in scale than the parent vortex form outside the central downdraft, in an annular region where both vertical velocity and its radial gradient are positive (Fig. 11, Rotunno 1984). Tornadogenesis occurs if one or more of these submesocyclone-scale vortices interact with the ground and consequently intensify into tornadoes (Rotunno 1986).
The tornadogenesis mechanism just described does not explain those tornadoes that develop near the mesocyclone axis or, in other words, those within one-celled mesocyclones. Moreover, it is unclear how often such a mechanism may be active: the percentage of all tornadoes that form from two-celled mesocyclones is unknown, as is the percentage of all mesocyclones that become two-celled some time during their life cycle (Trapp 1999). Yet another uncertainty regards the role(s) of ODs in tornadogenesis. Indeed, one may ask, “Is this downdraft a by-product of the tornadic vortex, or, is it necessary for the tornadogenesis process?” A proper evaluation of this question requires temporal and spatial data point/model gridpoint spacing that nominally can resolve a tornado; without such resolution, it is impossible to distinguish the cause from the effect. This criterion disqualifies most of the apparently relevant studies in the literature. Consider the relatively high-resolution modeling study of Grasso and Cotton (1995), which with its 111-m gridpoint spacing still does not really satisfy this criterion. This study appears to show a “tornado” that developed in absence of (or at least prior to) an OD. In Trapp and Fiedler (1995), the OD did not appear until after a tornado-like vortex was well formed, hence, the OD was an effect. On the other hand, Klemp and Rotunno (1983) and Wicker and Wilhelmson (1995) simulated submesocyclone-scale vortices that proceeded OD formation.
Further insight into occlusion downdrafts, two-celled mesocyclones, and attendant tornadogenesis awaits numerical modeling experiments performed with very high-resolution grids, and, high-resolution data collected for example by two Doppler On Wheels radars (Wurman et al. 1997) or by a millimeter-wavelength mobile radar (Bluestein et al. 1997).
4. Summary and conclusions
In a one-celled vortex, air rises throughout the vortex and may take the form of a jet that erupts out of the boundary layer. If this vortex undergoes vortex breakdown, the vertical jet terminates aloft at a stagnation point, downstream of which the vortex core broadens substantially, the core flow becomes turbulent, and recirculatory or downward motion develops along the central axis (Fig. 1a). In a two-celled vortex, downflow pervades the central axis of the vortex core, and terminates in low-level radial outflow that turns vertical in an annular updraft at an outer radius (Fig. 1b).
Observations of a downdraft near the central axis of a two-celled mesocyclonic vortex have been misinterpreted as evidence of vortex breakdown in tornadic mesocyclones. The argument that supports this statement is as follows. A supercritical vortex and a moderate-to-high swirl ratio are necessary for vortex breakdown. The supercriticality condition in tornadoes is provided through a boundary layer–erupting axial jet in the vortex core. Analytic model solutions suggest that an axial jet, driven by the boundary layer of the mesocyclone, unlikely forms in the mesocyclone core. Accordingly, the supercriticality condition unlikely is satisfied in mesocyclones, therefore precluding a mesocyclonic vortex breakdown.
Numerical model solutions demonstrate that an adverse vertical pressure gradient, induced by vertical vorticity that decreases in intensity with height, leads to the formation of an axial downdraft, hence, two-celled vortex, in absence of vortex breakdown. Two-celled mesocyclones, then, may form when low-level vertical vorticity, generated by the vertical tilting of horizontal baroclinic vorticity, exceeds that generated at midlevels through tilting and subsequent stretching of the horizontal vorticity associated with vertical shear of the ambient horizontal wind. Tornadogenesis may occur in a two-celled mesocyclonic vortex owing to an instability to which the vortex is susceptible.
Acknowledgments
Discussions with Drs. Rich Rotunno, Brian Fiedler, and Bob Davies-Jones were instrumental in helping the author “clarify” his thoughts on this topic. Additional comments by Dr. Chuck Doswell and two anonymous reviewers led to further improvements in the manuscript. The work was performed while the author was a visiting scientist with the Mesoscale and Microscale Meteorology Division of the National Center for Atmospheric Research. The National Center for Atmospheric Research is sponsored by the National Science Foundation.
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Schematic of (a) vortex breakdown in a tornado-like vortex and (b) a two-celled vortex (see text). Vectors indicate radial-vertical motion. Shading represents azimuthal or tangential motion, with higher windspeeds denoted by darker shading.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
Schematic of (a) vortex breakdown in a tornado-like vortex and (b) a two-celled vortex (see text). Vectors indicate radial-vertical motion. Shading represents azimuthal or tangential motion, with higher windspeeds denoted by darker shading.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
Schematic of (a) vortex breakdown in a tornado-like vortex and (b) a two-celled vortex (see text). Vectors indicate radial-vertical motion. Shading represents azimuthal or tangential motion, with higher windspeeds denoted by darker shading.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
Contours of (a) body force b, and of radial velocity u, tangential velocity υ, and vertical velocity w, at (b) t = 2.0, and (c) t = 4.5, for the free-slip experiment. Contour interval is indicated in the top right-hand corner of each plot, contour value begins at one-half the interval, and dashed lines indicate negative values.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
Contours of (a) body force b, and of radial velocity u, tangential velocity υ, and vertical velocity w, at (b) t = 2.0, and (c) t = 4.5, for the free-slip experiment. Contour interval is indicated in the top right-hand corner of each plot, contour value begins at one-half the interval, and dashed lines indicate negative values.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
Contours of (a) body force b, and of radial velocity u, tangential velocity υ, and vertical velocity w, at (b) t = 2.0, and (c) t = 4.5, for the free-slip experiment. Contour interval is indicated in the top right-hand corner of each plot, contour value begins at one-half the interval, and dashed lines indicate negative values.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
As in Fig. 2, except for the no-slip experiment, and for u, υ, and w at (a) t = 2.0, (b) t = 4.5, and (c) t = 6.0.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
As in Fig. 2, except for the no-slip experiment, and for u, υ, and w at (a) t = 2.0, (b) t = 4.5, and (c) t = 6.0.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
As in Fig. 2, except for the no-slip experiment, and for u, υ, and w at (a) t = 2.0, (b) t = 4.5, and (c) t = 6.0.
Citation: Monthly Weather Review 128, 3; 10.1175/1520-0493(2000)128<0888:ACOVBA>2.0.CO;2
In this context of axial jets within tornadoes or tornado-like vortices, upstream is in the −z (vertical) direction. With reference to the axial stagnation point, upstream is below the height of the stagnation point, and downstream is above the height of the stagnation point.
The swirl condition typically is imposed and external or ambient to the vortex core and will receive no further discussion.
The Rankine-combined vortex is characterized by a core of solid-body rotation [wherein tangential velocity (υ) ∼ radius (r)], surrounded by an outer region of potential flow (wherein υ ∼ 1/r).
These and the other variables have their traditional meanings.
