888 MONTHLY WEATHER REVIEW VOLUME 128 A Clari®cation of Vortex Breakdown and Tornadogenesis R. JEFFREY TRAPP National Severe Storms Laboratory, NOAA, and Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma 31 March 1999 and 9 August 1999 ABSTRACT Recent and past observations of a central downdraft within a mesocyclone have been misinterpreted as evidence of vortex breakdown on the mesocyclone scale. In this note, the literature is reviewed and model examples are provided in order to demonstrate that the mesocyclone dynamics likely precludes mesocyclonic vortex breakdown. It is shown that an adverse vertical pressure gradient, induced by vertical vorticity that decreases in intensity with height, leads to the formation of a central 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. Tor- nadogenesis occurs in a two-celled mesocyclonic vortex if an instability to which the vortex is susceptible is released. 1. Introduction 2. Two-celled vortices and vortex breakdown a. Background Vortex breakdown, a rather well-known phenomenon in ¯uid mechanics, has been documented in tornadoes Swirl ratio (S) is the ratio of the tangential velocity (Pauley and Snow 1988; see also Lugt 1989). The ex- at the outer edge of the updraft hole of a laboratory istence of vortex breakdown in tornadic mesocyclones tornado chamber (or the atmospheric analogue thereof), has been inferred from observations of a downdraft near to the average vertical velocity through such an updraft the central axis of the mesocyclonic vortex (Brandes hole (Davies-Jones 1973). The dependence on swirl ra- 1978; Wakimoto and Liu 1998); tornadogenesis was at- tio of tornadic vortex structure has been demonstrated tributed by these authors to the purported mesocyclonic in tornado chambers (e.g., Ward 1972; Church et al. vortex breakdown. Numerical models, however, have 1979) and numerical models thereof (e.g., Rotunno 1977; Howells et al. 1988). Indeed, the vortex ¯ow un- simulated the formation of such a central or axial down- dergoes a remarkable metamorphosis as S increases draft and, thus, of a two-celled vortex in the absence of (e.g., see Snow 1982; Davies-Jones 1986). Brie¯y, small vortex breakdown. values of S support a laminar, one-celled vortex: air rises The purpose of this note is to (i) demonstrate that throughout the vortex. Moderate swirl ratios support an vortex breakdown likely is precluded in mesocyclones, intense vortex, within which axial or vertical motions and (ii) provide a simple reinterpretation of these past now take the form of a jet that erupts out of the boundary and recent observations of central downdrafts and, layer. The vertical jet terminates aloft at a stagnation hence, of two-celled mesocyclonic vortices. These tasks point along the central axis, above which (i) the vortex are accomplished through a review of the literature and core broadens substantially, (ii) the ¯ow becomes tur- through a consideration of the observations and of an- bulent, and (iii) recirculatory or downward motion re- alytic and numerical model solutions (section 2). A the- sides along the axis (Fig. 1a): these are characteristics orized tornadogenesis process in two-celled mesocy- of the breakdown of a columnar vortex that terminates clones is recalled in section 3. Conclusions are given at a solid boundary (or end wall), in accordance with in section 4. the accepted de®nition 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 Corresponding author address: Dr. R. Jeffrey Trapp, NCAR/ ultimately, the vortex breakdown penetrates to the lower MMM, P.O. Box 3000, Boulder, CO 80307-3000. surface. This results in a broadly rotating, two-celled E-mail: [email protected] vortex with down¯ow along its central axis, terminating q 2000 American Meteorological Society Unauthenticated | Downloaded 09/24/21 07:11 PM UTC MARCH 2000 NOTES AND CORRESPONDENCE 889 mentioned, appears to be tied in some way to the ex- istence 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 con- trasts that of subcriticality that exists downstream of the breakdown point and allows for upstream and down- stream propagation of centrifugal waves (Benjamin 1962). According to Leibovich (1984), the general con- cept of critical ¯ow 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 supercrit- icality. b. Analytic models With the reasonable assumption that the velocity dis- tribution of a Rankine-combined vortex3 approximates that of a mesocyclone (and that of a tornado), the so- lution due to Bodewadt (1940) represents the effects of a low-level mesocyclone core and its boundary layer. FIG. 1. Schematic of (a) vortex breakdown in a tornado-like vortex Bodewadt examined a ¯uid that rotates as a solid-body and (b) a two-celled vortex (see text). Vectors indicate radial-vertical above a ®xed plate, and found that the resulting vertical motion. Shading represents azimuthal or tangential motion, with high- velocity distribution is independent of the radial dis- er windspeeds denoted by darker shading. tance 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 in low-level radial out¯ow that turns vertical in an an- throughout the mesocyclone core (of a few-kilometer nular updraft at an outer radius (Fig. 1b). radius) rather than of a narrow vertical jet on the me- The dynamics attendant with a rotating ¯ow above a socyclone axis (see Rotunno 1986, p. 430). Such a ver- rigid boundary (like the earth's surface) is crucial to tical jet does appear, however, in solutions of Burgraff development of an axial jet within simulated vortex et al.'s (1971) model of a potential vortex above a ®xed cores and, presumably, within tornadoes. Assume that plate; their model represents reasonably well a narrow the rotating ¯ow is in cyclostrophic balance, well above tornado and its boundary layer, since the presumed po- the rigid boundary and, hence, also above the boundary tential ¯ow of the tornadic vortex is large compared to layer. Within the boundary layer, the cyclostrophic bal- its core. ance condition is upset because friction reduces the tan- A swirling, boundary layer±erupting jet as in Burgraff gential wind and centrifugal force to zero at the ground; et al.'s solution (and in some tornadoes) is much more friction does not alter the radial pressure gradient force, likely to be supercritical than is a broadly rotating however. The consequential unbalanced inward pressure updraft as in Bodewadt's solution (and hence in some gradient drives a strong in¯ow 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 1 In this context of axial jets within tornadoes or tornado-like vor- tices, upstream is in the 2z (vertical) direction. With reference to depth, inward transport of mass and angular momentum the axial stagnation point, upstream is below the height of the stag- ceases as the ¯ow turns vertical, erupting into an intense nation point, and downstream is above the height of the stagnation axial jet (Burgraff et al. 1971); this rotating jet may point. break down under certain conditions. 2 The swirl condition typically is imposed and external or ambient to the vortex core and will receive no further discussion. Although vortex breakdown ``is still considered a ba- 3 The Rankine-combined vortex is characterized by a core of solid- sic, largely unexplained phenomenon in modern ¯uid body rotation [wherein tangential velocity (y) ; radius (r)], sur- dynamics'' (Rusak et al. 1998), its existence, as just rounded by an outer region of potential ¯ow (wherein y ; 1/r). Unauthenticated | Downloaded 09/24/21 07:11 PM UTC 890 MONTHLY WEATHER REVIEW VOLUME 128 low-level mesocyclone cores) (R. Davies-Jones 1999, sence of a boundary layer) is motivated by the presumed personal communication). Thus, by the discussion in relevance of the boundary layer to the scale of the vortex section 2a, it is correspondingly more unlikely for a being considered, as suggested by the model solutions mesocyclone to develop a vortex breakdown. Two-di- in section 2b: the boundary layer of a mesocyclone ap- mensional (2D), axisymmetric, numerical-model exper- parently has only a minor effect on the mesocyclone iments now are used to illustrate the formation of an core dynamics, whereas the boundary layer of a tornado axial downdraft and, thus, of a two-celled vortex in the has a profound effect on the tornado core dynamics. absence of vortex breakdown. 1) FREE-SLIP LOWER BOUNDARY CONDITION c. Numerical model results When a free-slip lower boundary condition is im- Consider the primitive equation version of the tor- posed, the model produces a subcritical vortex that be- nado-chamber numerical model developed by Fiedler comes two-celled in absence of a vortex breakdown (Fig. (1993), used recently by Trapp and Davies-Jones (1997); 2). Rapid development of an axial downdraft and con- the reader is referred to these papers for model details.